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Кодировка:

. . .

C. C.

A
4- ,

2001

517.98 22.16 95

. . . 4- ., . : - - , 2001. xii+354 c. ( ). ISBN 5861341036. . , Ё , , , , . - . , . .: 347. . . . .
1602080000-10 82(03)-2001

.

ISBN 5861341036

c . ., 2001 c . . . , 2001

1. § 1.1. § 1.3. ..................................... ....................... § 1.2.

viii xii 1 1 3 7 10 12 ................ 12 15 18 24 26 .......... ........ 26 29 32 35 38

.........................................

............................................

2. § 2.1. § 2.2. § 2.3.

............................. .........................

............................................

3. § 3.1. § 3.2.

§ 3.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 3.4. § 3.5. .........................

iv

§ 3.6. . . . . . . . . . . . . . . . . . . . . . . . . . . § 3.7. ......... ........

41 44 46 51 53 53 56 59 60 62 65 68 71 72 74 74 79 82 85 87 97 104

§ 3.8.

............................................

4. § 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 4.2. § 4.3. § 4.4. § 4.5. ...

..............................

....................................

.......................................... ......................... ......................... ...........

§ 4.6. § 4.7.

§ 4.8.

............................................

5. § 5.1. ......................

§ 5.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 5.3. ........................... .....

§ 5.4. § 5.5.

.......................... ...............

§ 5.6.

............................................

6. § 6.1. § 6.2. § 6.3. ....

v 106 106 111 114 119 122 125 129 131 131 134 138 141 147 150 155 158 . 158 165 172 175 179 187

.................................. ................................ ................

§ 6.4. § 6.5.

............................ ...............

§ 6.6.

............................................

7. § 7.1. § 7.2. ....................... ....................... ............... .

§ 7.3.

§ 7.4. § 7.5. § 7.6.

........

.........................

............................................

8. § 8.1. § 8.2.

.......

§ 8.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 8.4. ........................ ...........

§ 8.5.

............................................

vi 9. § 9.1. § 9.2.

190 190 193 196 201 207 213 218 220 220 223 226 228 230 232 234 236 243 251 260 272

......................

................................... ..............

§ 9.3. § 9.4.

....................................

§ 9.5. § 9.6. .................

............................................

10. Ё § 10.1. ........................... ................. ........ ..

§ 10.2.

§ 10.3.

§ 10.4. , § 10.5.

......................................... .......

§ 10.6. § 10.7. § 10.8. C (Q, R) § 10.9. § 10.10. D D

....................

..........................

................................... ..........................

§ 10.11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................................

11. § 11.1. § 11.2. ......

vii 274 274 276 278 280 281 283 288 290 294 300 303 323 327 345

.......................

§ 11.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 11.4. § 11.5. C (Q, C) § 11.6. § 11.7. C - .............

...................... ..................... ................... .......

§ 11.8.

§ 11.9. - C -

............................................

, . . , . , . , . , . , . . . , , (, ). . , , , . , . , . , , , . , . , , . , , , , , . , . . , . , . , . Ё , . . -

ix

. . . , . , . ., , . . . . , 1900 ., . . . , . , . , . , . , . ., . , , . . . . , . . . , (, . , , , , ). , , , : , . , , , , . : , . , . , , , .

x

, (. 1449). . , . . . . . , . . , . . , . . , . . , . . , . . . . . . 1948 . . . , , , 1974 ., . . , , . . , , , . .

. . , , . , . , . . . . , , .

. , , Kluwer Academic Publishers 1996 . , , , .

, , . C.

1

1.1. 1.1.1. . A B F A в B . F A B , , A B . 1.1.2. . F A в B dom F := D(F ) := {a A : ( b B ) (a, b) F } F , im F := R(F ) := {b B : ( a A) (a, b) F } F . 1.1.3. . (1) F F
-1

A B ,

:= {(b, a) B в A : (a, b) F }

B A, F . , F F -1 . (2) F A F A в A. (3) F A в B . F , a A (a, b1 ) F

2

. 1.

(a, b2 ) F , b1 = b2 . , U A IU := {(a, a) A2 : a U }, IU A A, U . U 2 U . F A в B A B , F dom F = A. IU A = U . IU . F A в B F : A B . , dom F = A im F B . im F = B : F A B . , F -1 B в A , F : A B . (4) . , F : A B B (ba )aA , a ba (a A), (ba ). , (a, b) F , b = ba . , . (5) F A в B U A. F (U в B ) U в B F U F U F |U . F (U ) := im F |U U F . . , F , a F (a) = b, F ({a}) = {b}. F (a) . , , . , F -1 (U ) U B F -1 U F . 1.1.4. . F A в B G C в D G F := {(a, d) A в D : ( b) (a, b) F & (b, d) G} F G. G F A D.

1.2.

3

1.1.5. . , , , 1.1.4 , B = C . 1.1.6. F , F F -1 = I im . . F F -1 I imF . , F |dom F

F

1.1.7. F A в B , G B в C U A. G F A в C G F (U ) = G(F (U )). 1.1.8. F A в B , G B в C , H C в D. H (G F ) A в D (H G) F A в D . 1.1.9. . 1.1.8 H G F . 1.1.10. F, G, H H GF =
(b,c)G

. F
-1

(b) в H (c).

(a, d) H G F ( (b, c) G) (c, d) H & (a, b) F ( (b, c) G) a F -1 (b) & d H (c) 1.1.11. . 1.1.10 , , , ( , 1.1.1). . ( ) , , . 1.1.12. G F GF =
b imF

F

-1

(b) в G(b).

1.1.10 : H := G, G := I

imF

F := F .

4

. 1. 1.2.

1.2.1. . X , . . X 2 . IX , , -1 IX , , = -1 . 1.2.2. . . . . X , X , (X, ) x y y (x). : X , x y x y y x . . , . . . . 1.2.3. . (1) ; X0 X 0 := X0 в X0 . (2) () X , -1 () X . -1 (). (3) f : X Y Y . X : f -1 f . 1.1.10 f
-1

f =
(y1 ,y2 )

f

-1

(y1 ) в f

-1

(y2 ).

, (x1 , x2 ) f
-1

f (f (x1 ), f (x2 )) .

, , f -1 f , f . , .

1.2.

5

. , , , : f -1 f = f -1 IY f . (4) X X . : X 2X (x) := (x) ( 2X X , P (X )). X := X/ := im . , , ( , . .). , X . (x) x. , =
-1

=
xX

-1

(x) в

-1

(x).

f : X Y . f f X , . . f : X Y , f = f , f -1 f . (5) (X, ) (Y , ) . f : X Y (. . x y f (x) f (y )) , f -1 f . 1.2.4. . (X, ) U X . x X U , U -1 (x). : x U . , x . x X U , x U -1 . : x U . , x . 1.2.5. . , (). , . 1.2.6. . x U , x U x U . U .

6

. 1.

1.2.7. (U ) U (X, ). , , x X U . , -, x (U ), -, (x) U = {x}. 1.2.8. . 1.2.7 . 1.2.9. . x X U X , x U . x = supX U , , x = sup U . ( ) U inf U , , inf X U . 1.2.10. . x (X, ) U X , (x) U = {x}. U . 1.2.11. . . , , , . 1.2.12. . X , x1 , x2 X x1 x2 := sup{x1 , x2 } x1 x2 := inf {x1 , x2 }. 1.2.13. . X , X . 1.2.14. , . 1.2.15. . (X, ) , X 2 = -1 , . -

1.3.

7

. , , . 1.2.16. . X () X . ( ) N X () . ( , N := {1, 2, 3 . . . }.) 1.2.17. , . 1.2.18. . 1.2.17 , X X X . 1.2.19. . (X, ) X 2 = -1 . X . X0 X , X0 X . , (. . ). 1.2.20. . . -

1.2.21. . , . 1.3. 1.3.1. . X B 2X . B ( X ), B 2X X .

8

. 1.

1.3.2. B 2X , (1) B = , B ; (2) B1 , B2 B ( B B ) B B1 B2 . 1.3.3. . F 2X ( X ), F B ( X ), . . F = fil B := {C 2X : ( B B ) B C }. , B . . 1.3.4. F , (1) F = , F (2) A F , A B (3) A1 , A2 F F F B 2X ; X B F; A1 A2 F .

1.3.5. . (1) F X в Y B 2X . F (B ) := {F (B ) : B B }. , F (B ) . , F (B ) := fil F (B ). F X B dom F = B F , F (F ) Y . F F . , F : X Y B X , F (F ) Y . (2) (X, ) . , B := { (x) : x X } . F : X Y , fil F (B ) F . (X , ) F : X Y Y . F F , F ( ) F . ( ) G : X X (x)xX (X, ) , F = F G, F F ( : F F ). .

1.3.

9

1.3.6. . F (X ) X . F1 , F2 F (X ) F1 F2 , , F1 F2 F1 F2 ( F2 F1 F2 F1 ). 1.3.7. F (X ) . 1.3.8. N F (X ). N F0 := sup N . F0 = {F : F N }. , F0 . , F0 , N , F0 = . A F0 B A, / , F N , A F , : B F F0 . A1 , A2 F0 , F N , A1 , A2 F , N . 1.3.4, A1 A2 F F0 . 1.3.9. . F (X ) X . 1.3.10. . 1.3.8 , , . 1.2.20. 1.3.11. F , A X A F , X \ A F . : A F B := X \ A F . , A = B = . F1 := {C 2X : A C F }. A F F1 B F1 F1 = . 1.3.4 (2) 1.3.4 (3). , F1 . F1 F . F , F1 = F . : B F B F. : F1 F (X ) F1 F . A F1 A F , X \ A F . X \ A F1 , . . = A (X \ A) F1 , .

10

. 1.

1.3.12. f X Y F X , f (F ) Y .

1.3.13. X := XF0 := {F F (X ) : F F0 } F0 F (X ). X . , F0 , {X } X . , X : sup = inf X = {X } inf = sup X = F0 . 1.2.17 1.3.8 F1 F2 F1 , F2 X . F := {A1 A2 : A1 F1 , A2 F2 }. , F F0 F F1 , F F2 . F = F1 F2 , F . F = F . , (B1 , B2 F B1 B2 F ). , C A1 A2 , A1 F1 A2 F2 , C = {A1 A2 } C = (A1 C ) (A2 C ). A1 C F1 , A2 C F2 , : C F . 1.3.4 .
1.1. , - - . 1.2. [0, 1] [0, 1]? [0, 2]? 1.3. : , , RM в RN M , N . 1.4. R, S, T : (R S )
-1

=R

-1

S

-1

;

(R S )

-1

=R

-1

S

-1

;

(R S ) T = (R T ) (S T ); (R S ) T (R T ) (S T );

R (S T ) = (R S ) (R T ); R (S T ) ( R S ) ( R T ) .

1.5. X X в X . , X =

.

1.6. XA = B AX = B X , . 1.7. . 1.8. ? ?

11

1.9. ( ). 1.10. - ? . ? 1.11. F , X . , F : F IX . F () , , . . 1.12. X, Y M (X, Y ) X Y (?). , (1) (M (X, Y ) ) (Y ); (2) (M (X, Y ) ) (Y ). 1.13. , X, Y , Z : (1) M (X, Y в Z ) M (X, Y ) в M (Y , Z ); (2) M (X в Y , Z ) M (X, M (Y , Z )). 1.14. ? 1.15. ? 1.16. f X Y . , Y f X . 1.17. , , , . 1.18. , . 1.19. A N, . x, y s := RN x A y := ( A A ) x|A = y |A . R := RN /A . t R t , t(n) := t (n N). , R \ {t : t R} = . R . R R?

2

2.1. 2.1.1. . , , . X A (X, +) A X , · : A в X X . . : X A (X, A, +, ·) . 2.1.2. . R C . F. , R ( ) C. 2.1.3. . F . X F ( F). F , X . R , C . : (X, F, +, ·), (X, R, +, ·) (X, C, +, ·). , , ґ , X . 2.1.4. . (1) F F. (2) (X, F, +, ·) . -

2.1.

13

(X, F, +, · ), · : (, x) x s F x X , . X X . F := R X X . (3) (X0 , F, +, ·) (X, F, +, ·), X0 X X0 F в X0 X . X0 X . , , X0 X . , X X 0. X , , , . (4) (X ) F. , , X := X , . . x : X , x := x( ) X ( , = ). X : (x1 + x2 )( ) := x1 ( ) + x2 ( ) (x1 , x2 X , ); ( · x)( ) := · x( ) (x X , F, ) (, , · x : x x). X F (X ) . := {1, 2, . . . , N } X1 в X2 в . . . в XN := X . , X = X , X := X . := {1, 2, . . . , N }, X N := X . (5) (X ) F. X0 := X , . . X := X , x0 , ( , x0 ) 0 , x0 ( \ 0 ) 0. ,

14

. 2.

X0 X . (X ) (X ) . (6) (X, F, +, ·) (X0 , F, +, ·) X . X0 := {(x1 , x2 ) X 2 : x1 - x2 X0 }. X0 X . X := X/ X0 : X X . X x1 + x2 := (
-1

(x1 ) +
-1

-1

(x2 )) (x1 , x2 X );

x := (

(x)) (x X , F). F

, , S1 , S2 X , F , S1 + S2 := +{S1 в S2 }; S1 := · ( в S1 ); S1 := {}S1 .

X F. - X X0 X/X0 . 2.1.5. X Lat(X ) X . Lat(X ) . , inf Lat(X ) = 0 sup Lat(X ) = X . , . 1.2.17, . 2.1.6. . X1 , X2 Lat(X ) X1 X2 = X1 + X2 . , E Lat(X ) inf E = {X0 : X0 E }. E , sup E = {X0 : X0 E }.

2.2.

15

2.1.7. . X1 X2 X X () ( : X = X1 X2 ), X1 X2 = 0 X1 X2 = X . X2 () X1 , X1 () X2 . 2.1.8. . X1 X . E := {X0 Lat(X ) : X0 X1 = 0}. , 0 E E0 E , 2.1.6, X1 sup E0 = 0, . . sup E0 E . , E , 1.2.20 E X2 . x X \ (X1 + X2 ), (X2 + {x : F}) X1 = 0. , F x1 X1 , x2 x2 + x = x1 , x X1 + X2 , , x1 = x2 = 0, X1 X2 = 0. , X2 + {x : X2 . , x x = 0. X1 X2 = X1 + X2 = 2.2. 2.2.1. . X , Y F. T X в Y , T X в Y . T : X Y , , ( , ). T S X в Y dom S = X , : T ( X Y ) S X Y , S . 2.2.2. T : X Y , T (1 x1 + 2 x2 ) = 1 T x1 + 2 T x2 (1 , 2 F; x1 , x2 X ). = = X X 0. 0. .
2

F} = X 2

16

. 2.

2.2.3. L (X, Y ) X Y Y X. 2.2.4. . L (X, F) X , X # := L (X, F) () . X X . F, , . . , F = R - , , . 2.2.5. . T L (X, Y ) () , T -1 L (Y , X ). 2.2.6. . X Y () X Y , X Y . 2.2.7. X Y , T L (X, Y ) S L (Y , X ) , S T = IX T S = IY . S = T -1 T = S -1 . 2.2.8. . X, Y , Z , T L (X, Y ) S L (Y , Z ). , S T L (X, Z ). S T S T . , (S, T ) S T , , : L (Y , Z ) в L (X, Y ) L (X, Z ). , E L (Y , Z ), T L (X, Y ), E T := (E в {T }). 2.2.9. . (1) T , T -1 . (2) X1 X X2 , X2 X/X1 . , : X X/X1 ,

2.2.

17

X2 , . . x2 (x2 ), x2 X2 , . (3) X := X (X ) . Pr : X X , Pr x := x , (= ). , Pr : Pr L (X , X ). , L (X ) := L (X , X ), X X , X := X , X := 0 = X := X . (4) X := X1 X2 . +-1 X X1 в X2 , PX1 := PX1 ||X2 := Pr1 (+-1 ), PX2 := PX2 ||X1 := Pr2 (+-1 ), X X . PX1 X X1 X2 , PX2 PX1 . , PX1 PX2 , PX2 X X2 X1 . , 2 PX1 + PX2 = IX . , PX1 := PX1 PX1 = PX1 , . . . , P L (X ) P (X ) P -1 (0). T L (X ), PX1 T PX1 = T PX1 , T (X1 ) X1 , . . X1 T . T PX1 = PX1 T , X1 , X2 T . , X = X1 X2 T . T X1 T1 L (X1 ). T1 T X1 . T2 L (X2 ) T X2 , T T T1 0 0 T2 .

, x X1 X2 - x1 X1 , x2 X2 , x1 = PrX1 x, x2 = PrX2 x. , x, . . - T1 x1 , T2 x2 (,

18

. 2.

, T x1 , T x2 ), T x. , T X1 в X2 X1 в X2 , x1 x2 T1 0 0 T2 x1 x2 .

T L (X1 X2 , Y1 Y2 ). (5) E X , eE e e = 0, e F (e E ), , e = 0 e E . E , E . X ( ) X . . X , X . X dim X . X (F) , dim X . X1 X , X/X1 X1 codim X1 . X = X1 X2 , codim X1 = dim X2 dim X = dim X1 + codim X1 . 2.3. 2.3.1. . T L (X, Y ) : ker T := T -1 (0) , coker T := Y / im T , coim T := X/ ker T T . T , ker T = 0. T , im T = Y . 2.3.2. , .

2.3.

19

2.3.3. . . , . , 1 -Y X @ 2 4 3@ R @? V - W 5 , 1 L (X, Y ), 2 L (Y , W ), 3 L (X, W ), 4 L (V , Y ) 5 L (V , W ), 2 1 = 3 5 = 2 4 . 2.3.4. . X Y Z ( Y ) , ker S = im T . . . . Xk-1 Xk Xk+1 . . . Xk , Xk-1 Xk Xk+1 ( ). , ( , , , ). 2.3.5. . (1) X Y Z , . . S T = 0. . (2) 0 X Y , T . ( 0 X , , L (0, X ) (. 2.1.4 (3)).) (3) X Y 0 , T . (, Y 0 L (Y , 0).) (4) T L (X, Y ) T , 0 X Y 0 .
T T T S T S

20

. 2.

(5) X0 X . : X0 X () : x0 := x0 x0 X0 . X/X0 - : X X/X0 . 0 X0 X X/X0 0 . ( , , .) . , , , 0XY Z0 , . Y0 := im T , , , , : 0 -X ? - Y0 T -Y -Z ? ? -Y - Y / Y0 S -0
T S

0

-0

, , - . (6) T L (X, Y ) . 0 ker T X Y coker T 0, T . 2.3.6. . T T ( T T0 ), X
0 0 T

-X @ T T0@ ? R @ Y

. . T0 = T , : X0 X

.

2.3.

21

2.3.7. X, Y X0 X . T0 L (X0 , Y ) T L (X, Y ). T := T0 PX0 , PX0 X0 . 2.3.8. XA = B . X, Y , Z ; A L (X, Y ), B L (X, Z ). X A -Y @ X B@ ? R @ Z X L (Y , Z ) , ker A ker B . : , B = X A ker A ker B , . : X := B A-1 . , x X X A(x) = B (A-1 A)x = B (x + ker A) = B x. , X0 := X |im A . X . y im A z1 , z2 X (y ). z1 = B x1 , z2 = B x2 , Ax1 = Ax2 = y . B (x1 - x2 ) = 0. , z1 = z2 . 2.3.7, - X X0 Y . 2.3.9. . 2.3.8 A , X .

2.3.10. . 2.3.8 2.3.9. 2.3.11. T () , T , . . :

22 coim T 6 X T .

. 2. Tim T T ? -Y

2.3.12. X f0 , f1 , . . . , fN X # . f0 f1 , . . . , fN , ker f0 N ker fj . j =1 (f1 , . . . , fN ) : X F
N

,

(f1 , . . . , fN )x := (f1 (x), . . . , fN (x)). , ker(f1 , . . . , fN ) = N ker fj . 2.3.8 j =1 X
(f1 ,...,fN )

-F @ @ f0@

N

@ R @? F F
N#

, .

2.3.13. AX = B . X, Y , Z ; A L (Y , X ), B L (Z, X ). A X Y

6 I @ @ X B@ Z X L (Z, Y ) , im A im B .

2.3.

23

: im B = B (Z ) = A(X (Z )) A(Y ) = im A. : Y0 ker A Y A0 := A|Y0 . A0 Y0 im A. X := A-1 B , , . 0 2.3.14. . 2.3.13 A , X . -

2.3.15. . 2.3.8 2.3.13 . , . 2.3.16. . S L (Y , Z ) T L (X, Y ). , , 1 , . . . , 6 , :

()
1 2 3 0 ker T - ker S T - ker S - 3 4 5 - coker T - coker S T - coker S 0

.

24

. 2.

2.1. , . ? 2.2. Z2 . 2.3. . 2.4. f : R R f (x + y ) = f (x) + f (y ) (x, y R). f ? 2.5. , , , . 2.6. X X0 X00 . , X/X00 (X/X0 )/(X00 /X0 ) . 2.7. : x
##

:x

#

x|x

#

(x X , x

#

X # ).

, X X ## . 2.8. , , . .
##

(X ) = X

##

dim X < +.

2.9. ? 2.10. ? 2.11. T , T n-1 = 0 T n = 0 - n. , T 0 , T , . . . , T n-1 . 2.12. , . 2.13. XA = B AX = B ( X ). 2.14. ? 2.15. , .

25

2.16. (xe )eE (x# )eE , : e xe | x xe | x
# e # e

=1

(e E );

=0

(e, e E, e = e ).

2.17. (x# )eE e (xe )eE , : xe | x xe | x
# e # e

=1

(e E );

=0

(e, e E, e = e ).

2.18. . 2.19. W - X - Y - Z W - X - Y - Z , , ker = T (ker ) T -1 (im ) = im . . ,
T T

3

3.1. 3.1.1. . F 2 , U . U ( U ( )), (1 , 2 ) 3.1.2. . (1) (). ( , () .) (2) := F 2 - . (3) := R2 , - X X. (4) := R2 , - + . , K , K + K K K K R+ . R2 \ 0- () + , R+ в 0- . ( R+ := {t R : t 0}.) (5) := {(1 , 2 ) F 2 : 1 + 2 = 1}. - . X0 1 U + 2 U U.

3.1.

27

X x X , x + X0 := {x} + X0 X . , L X x L, L - x := L + {-x} X . (6) := {(1 , 2 ) F 2 : |1 | + |2 | 1}. - . (7) := {(, 0) F 2 : || 1}. - ( F := R ; , ). (8) := {(1 , 2 ) R2 : 1 0, 2 0, 1 + 2 = 1}. - . (9) := {(1 , 2 ) R2 : 1 + 2 1}, + - . , . (10) F 2 X F X ( ). , 3.1.2 (1)3.1.2 (9) -. 3.1.3. X E - X . im E } ( ). , , im E ( ), {U : U im {U : U E } ( ).

3.1.4. . 3.1.3, , , - , , . Y 3.1.5. X Y , U X V -. U в V ( ).

U V , U в V = . u1 , u2 U v1 , v2 V , (1 , 2 ) . 1 u1 + 2 u2 U , 1 v2 + 2 v1 V . , (1 u1 + 2 u2 , 1 v1 + 2 v2 ) U в V . 3.1.6. . X , Y T XвY . F 2 . T ( ), T -.

28

. 3.

3.1.7. . - ( ) , . , . . : , (. 3.4.2). 3.1.8. T X в Y 1 -, U X 2 -. 2 1 , T (U ) ( 2 ). y1 , y2 T (U ), x1 , x2 U (x1 , y1 ) T (x2 , y2 ) T . (1 , 2 ) 2 (1 , 2 ) 1 , , 1 (x1 , y1 ) + 2 (x2 , y2 ) T . , 1 y1 + 2 y2 T (U ). 3.1.9. - -. F X в V G W в Y F, G ( ). (x1 , y1 ) G F ( v1 ) (x2 , y2 ) G F ( v2 ) (x1 , v1 ) F & (v1 , y1 ) G; (x2 , v2 ) F & (v2 , y2 ) G.

1 , 2 , (1 , 2 ) , , . 3.1.10. U , V X U , V ( ) , F U + V ( ). 3.1.5, 3.1.8 3.1.9. F 2,

3.1.11. . X , F 2 U X . H (U ) := {V X : V ( ), V U } - U . 3.1.12. : (1) H (U ) ( ); (2) H (U ) -, U ; (3) U1 U2 H (U1 ) H (U2 ); (4) U ( ) U = H (U ); (5) H (H (U )) = H (U ).

3.2. 3.1.13. : H (U ) = {H (U0 ) : U0 U, U
0

29

}.

V , . U0 U , , 3.1.12 (3), H (U0 ) H (U ), H (U ) V . 3.1.12 (2) (, , ) , V ( ). 3.1.3 , H (U0 ) H (U1 ) H (U0 U1 ). 3.1.14. . , - - . , ( ) -. , := {(1 , 2 ) R2 : 1 + 2 = 1} + H (U ) co(U ). HF 2 (U ) L (U ) lin(U ), U = , , L () := 0. L (U ) U ( L (X ) X ). , . . , , . .
N

co({x1 , . . . , xN }) =
k=1

k xk : k 0, 1 + . . . +

N

=1 .

3.2. 3.2.1. . (X, R, +, · ) . , , X . , , X 2 . X . ( (X, R, +, ·, ), , .) 3.2.2. X . (0) . (x) = x + (0) x X .

30

. 3.

(0) 3.1.3. , (x, y ) = (x, x) + (0, y - x) (x, y ) y - x (0). 3.2.3. K X .

:= {(x, y ) X 2 : y - x K }. , , K (0). , , K (-K ) = 0. , 0 K IX K + K K . -1 = {(x, y ) X 2 : x - y K }. , -1 IX K (-K ) = 0. , . (x1 , y1 ), (x2 , y2 ) 1 , 2 R+ . 1 y1 + 2 y2 - (1 x1 + 2 x2 ) = 1 (y1 - x1 ) + 2 (y2 - x2 ) 1 K + 2 K K. 3.2.4. . K , K (-K ) = 0. 3.2.5. . 3.2.2 3.2.3 . . () X (X, X+ ), X+ . 3.2.6. . (1) R R+ := (R+ ) , . (2) X X+ . X0 X , , X0 X , X0 X+ . X0 . (3) X Y () . T L (X, Y ) ( T 0), T (X+ ) Y+ . L+ (X, Y ).

3.2.

31

L+ (X, Y ) Lr (X, Y ). Lr (X, Y ) . 3.2.7. . , . 3.2.8. . , , K -, . 3.2.9. K - . x U . -x -U . , 3.2.8 sup(-U ). -x sup(-U ). , - sup(-U ) = inf U . 3.2.10. K - U V sup(U + V ) = sup U + sup V . , U V , . . , sup(U + V ) = sup{sup(u + V ) : u U } = = sup{u + sup V : u U } = sup V + sup{u : u U } = = sup V + sup U. 3.2.11. . 3.2.10 , . : sup U = sup U R+ . 3.2.12. . x x+ := x 0 x, x- := (-x)+ , |x| := x (-x) x.

32

. 3.

3.2.13. x y x + y = x y + x y. x + y - x y = x + y + (-x) (-y ) = y x 3.2.14. x = x+ - x- ; |x| = x+ + x- . 3.2.13 y := 0. , |x| = x (-x) = -x + (2x) 0 = -x + 2x+ = (x+ - x- ) + 2x+ = x+ + x- . 3.2.15. . x, y X [0, x + y ] = [0, x] + [0, y ]. ( , [u, v ] := (u) -1 (v ) () .) [0, x] + [0, y ] [0, x + y ] . 0 z x + y , z1 := z x. , z1 [0, x]. z2 := z - z1 . z2 0. z2 = z - z x = z + (-z ) (-x) = 0 (z - x) 0 (x + y - x) = 0 y = y . 3.2.16. . X , Y K -. Lr (X, Y ) L+ (X, Y ) K -. 3.3. 3.3.1. . (1) X B ([0, 1], R) [0, 1], X0 := C ([0, 1], R) X , . Y := X0 X0 , X Y (. 3.2.6 (1) 3.2.6 (2)). T0 : X0 Y T L+ (X, Y ). T , E X0 supX0 E , X0 . , supX0 E = T supX E , supX E E X . , Y K -.

3.3.

33

(2) s := RN , . , , c s, . , f0 : c R, f0 (x) := lim x(n), s. , f s# , f 0 f f0 . x0 (n) := n xk (n) := k n k , n N. , f0 (xk ) = k . , f (x0 ) f (xk ) 0, x0 xk 0. . 3.3.2. . X0 X X+ ( X ), X0 + X+ = X . 3.3.3. X0 X , x X x0 , x0 X0 , x0 x x0 . 3.3.4. . X , X0 X Y K -. T0 L+ (X0 , Y ) T L+ (X, Y ). I. X := X0 X1 , X1 , X1 := {x : R}. X0 T0 , U := {T0 x0 : x0 X0 , x0 x} , , y := inf U . T x := {T0 x0 + y : x = x0 + x, x0 X0 , R}. , T , T T0 dom T = X . T . x = x0 + x x 0, = 0 . > 0, x -x0 /. , -T0 x0 / y , . . T x Y+ . < 0 x -x0 /. , y -T0 x0 / T x = T0 x0 + y Y+ . II. E S X в Y , S T0 S (X+ ) Y+ . 3.1.3 E , E T .

34

. 3.

x X \ dom T , I X := dom T X1 , X0 := dom T , T0 := T X1 := {x : R}. T . , T . 3.3.5. . Y := R 3.3.4 . 3.3.6. . x , [0, x] = [0, 1]x. 3.3.7. (X, X+ ) , X = X+ - X+ . f T+ T |X T X := X+ - X+ . X # . , ker f X f = 0. f [0, T ], . . [0, 1] T + f = T . = 0, 2T [0, T ]. T = 0 f = 0. T (x0 ) = 0 - x0 X , = 1 f = 0. X , X X T0 T X , -

3.3.8. . X0 . T0 .

3.3.4. I. T . , T [0, T ] [0, 1] x0 X0 T (x0 ) = T (x0 ) (T - T )(x0 ) = (1 - )T (x0 ). : T (x) inf {T (x0 ) : x0 x, x0 X0 } = T (x); (T - T )(x) inf {(T - T )(x0 ) : x0 x, x0 X0 } = (1 - )T (x). , T = T [0, T ] [0, 1]T . . , T . II. E , 3.3.4. Ed , S E , S |dom S dom S . Ed . 1.2.19 E0 Ed . S := {S0 :

3.4.

35

S0 E0 }. , S E . S , . S (dom S )# , 0 S (x0 ) S (x0 ) x0 (dom S )+ . S (x0 ) = 0 x0 , S = 0S , . S (x0 ) = 0 x0 (dom S )+ , S0 E0 S0 (x0 ) = S (x0 ). S0 : S (x ) = S (x ) x dom S0 . = S (x0 )/S (x0 ), . . S0 . E0 , : S = S . 3.4. 3.4.1. . R· R· +. (+) := + ( R+ ), + + x := x + (+) := + (x R· ). 3.4.2. . f : X R· . -

epi f := {(x, t) X в R : t f (x)} f , dom f := {x X : f (x) < +} f . 3.4.3. dom f f , . . , : X R· f X в R X R. dom f = X , f : X R, R· .

3.4.4. . X . f : X R· , epi f .

36

. 3.

3.4.5. f : X R· , , . . f (1 x1 + 2 x2 ) 1 f (x1 ) + 2 f (x2 ), 1 , 2 0, 1 + 2 = 1 x1 , x2 X . : 1 , 2 0, 1 + 2 = 1 x1 , x2 dom f , . x1 , x2 dom f . (x1 , f (x1 )) epi f (x2 , f (x2 )) epi f . , 3.1.2 (8), 1 (x1 , f (x1 )) + 2 (x2 , f (x2 )) epi f . : f : X R· (x1 , t1 ) epi f , (x2 , t2 ) epi f , . . t1 f (x1 ) t2 f (x2 ) ( dom f = f (x) = + (x X ) epi f = ). , , 1 , 2 0, 1 + 2 = 1 (1 x1 + 2 x2 , 1 t1 + 2 t2 ) epi f . 3.4.6. . p : X R· , epi p . 3.4.7. dom p = 0 : (1) p ; (2) p , ; . . p(x) = p(x) 0 x dom p; (3) 1 , 2 R+ x1 , x2 X p(1 x1 + 2 x2 ) 1 p(x1 ) + 2 p(x2 ); (4) p , : p(x1 + x2 ) p(x1 ) + p(x2 ) x1 , x2 X . 3.4.8. . (1) , . (2) U X . (U )(x) := 0, x U, +, x U.

(U ) : X R· U . , (U ) . U ,

3.4.

37

(U ) . U , (U ) . (3) ( ) ( , . . (R· )X ) . . (4) (. . ) . . 3.4.9. . X , U V X . , U V , n N, V nU . U ( X ), U X , . . X = nN nU . 3.4.10. T X в Y , im T = Y . U ( X ), T (U ) ( Y ). Y = T (X ) = T (nN nU ) = nN T (nU ) = nN nT (U ) 3.4.11. . U X . x U core U U ( U ), U - x X . 3.4.12. f : X R· x core dom f . h X f (x)(h) := lim
0

f (x + h) - f (x) f (x + h) - f (x) = inf . >0

f (x) : h f (x)h f (x) : X R. () := f (x + h). 3.4.8 (4) : R R· . 0 core dom . (() - (0))/ ( > 0) , . . (0)(1). f (x)(h) = (0)(1).

38

. 3. > 0 h H f (x)( h) = inf = inf f (x + h) - f (x) =

f (x + h) - f (x) = f (x)(h).

, h1 , h2 X f (x)(h1 + h2 ) = 2 lim
0 1 f x + 2 (h1 + h2 ) - f (x) =

= 2 lim
0

f

1 2

1 (x + h1 ) + 2 (x + h2 ) - f (x)

lim
0

f (x + h1 ) - f (x) f (x + h2 ) - f (x) + lim = 0 = f (x)(h1 ) + f (x)(h2 ).

3.4.7 . 3.5.

3.5.1. . X , f : X R· x dom f . x (f ) := {l X # : ( y X ) l(y ) - l(x) f (y ) - f (x)} f x. 3.5.2. . (1) p : X R· . p (p) := 0 (p). (p) = {l X # : ( x X ) l(x) p(x)}; x (p) = {l (p) : l(x) = p(x)}. (2) l X # . (l) = x (l) = {l}. (3) X0 X . ( (X0 )) = {l X
#

: ker l X0 }.

3.5.

39

(4) f : X R· x core dom f .

x (f ) = (f (x)). 3.5.3. . T L (X, Y ) , f : Y R· , x X , T x core dom f . x (f T ) = T x (f ) T . 3.4.10 , x core dom f . 3.5.2 (4), x (f T ) = ((f T ) (x)). , h X (f T ) (x)(h) = lim
0

(f T )(x + h) - (f T )(x) =

= lim
0

f (T x + T h) - f (T x) = f (T x)(T h).

p := f (T x). 3.5.2 (4) , , 3.4.12, p , : (p) = (f (T x)) = T x (f ); (p T ) = ((f T ) (x)) = x (f T ). , (p T ) = (p) T . l (p) T , . . l = l1 T , l1 (p), l1 (y ) p(y ) y Y . , l(x) l1 (T x) p(T x) = p T (x) x X , . . l (p T ). , (p) T (p T ). l (p T ). T x = 0, l(x) p(T x) = p(0) = 0, . . l(x) 0. -x. l(x) = 0. , ker l ker T . , 2.3.8, l = l1 T l1 Y # . Y0 := T (X ) Y0 Y , , l1 (p ). , (p ) (p) ,

40

. 3.

l2 (p) l1 = l2 . l = l1 T = l1 T = l2 T = l2 T , . . l (p) T . , , (p ) (p) . l0 (p ) Y0 := Y0 в R Y := Y в R T0 : (y0 , t) t - l0 (y0 ). Y Y+ := epi p. , -, Y0 (y , t) = (0, t - p(y )) + (y , p(y )) (y Y , t R). -, (y0 , t) Y0 Y+ , 3.4.2, t p(y0 ) , , T0 (y0 , t) = t - l0 (y0 ) 0, . . T0 Y0 . 3.3.4 T Y, T0 . l(y ) := T (-y , 0) y Y . , l = l0 . , T (0, t) = T0 (0, t) = t. , 0 T (y , p(y )) = p(y ) - l(y ), . . l (p). 3.5.4. . 3.5.3 , . , (p ) (p) : , , , . 3.5.5. . X , X0 X p : X R . () : (p + (X0 )) = (p) + ( (X0 )). . l (p + (X0 )). l (p ), X0 X . 3.5.3, l (p) , . . l1 (p) l = l1 . l2 := l - l1 . l2 = (l - l1 ) = l - l1 = 0, . . ker l2 X0 . 3.5.2 (3), , l2 ( (X0 )).

3.6.

41

3.5.6. . f : X R· x core dom f . x (f ) = . p := f (x), : 0 X . , 0 (p ), . . (p ) = . 3.5.3, (p) = ( = (p) = (p )). 3.5.2 (4). 3.5.7. . f , f : X R·
1 2

x core dom f1 core dom f2 . x (f1 + f2 ) = x (f1 ) + x (f2 ). p1 := f1 (x) p2 := f2 (x). x1 , x2 X p(x1 , x2 ) := p1 (x1 ) + p2 (x2 ) (x1 ) := (x1 , x1 ). 3.5.2 (4) 3.5.3, : x (f1 + f2 ) = (p1 + p2 ) = (p ) = = (p) = (p1 ) + (p2 ) = x (f1 ) + x (f2 ). 3.5.8. . 3.5.6 . , . , 3.5.6, , (p T ) = (p) T , . pT (y ) := inf {p(y + T x) - l(x) : x X }, l (p) 3.5.3. , pT l1 (pT ) l = l1 T . , ( ) . 3.6. 3.6.1. . X seg X 2 в X , seg(x1 , x2 ) := {1 x1 + 2 x2 : 1 , 2 > 0, 1 + 2 = 1}.

42

. 3.

, , V X segV seg V 2 . U , V , V , seg-1 (U ) U 2 . V . x V V , {x} V . V ext(V ). 3.6.2. U V , v1 , v2 V , 1 , 2 > 0, 1 + 2 = 1 1 v1 + 2 v2 V , v1 U v2 U . 3.6.3. . (1) p : X R· x X dom p. x (p) (p). , 1 , 2 > 0 1 +2 = 1 , 1 l1 + 2 l2 x (p) l1 , l2 (p), 0 = p(x) - (1 l1 (x) + 2 l2 (x)) = 1 (p(x) - l1 (x)) + 2 (p(x) - l2 (x)) 0. , p(x) - l1 (x) 0 p(x) - l2 (x) 0. , l1 x (p) l2 x (p). (2) U V , , V W . U W . (3) X . x X+ , {x : R+ } X+ . : 0 y x. x = 1/2 (2y ) + 1/2 (2(x - y )). 3.6.2, 2y = x 2(x - y ) = x , R+ . , 2x = ( + )x. x = 0, . x = 0, /2 [0, 1] , , [0, x] [0, 1]x. . : [0, x] = [0, 1]x 0; 1 , 2 > 0, 1 + 2 = 1 y1 , y2 X+ x = 1 y1 + 2 y2 . = 0, 1 y1 [0, x] 2 y2 [0, x] , , y1 y2 . > 0, (1 /)y1 = tx t [0, 1]. , (2 /)y2 = (1 - t)x. (4) U . V U U , U \ V .

3.6. {x}

43

x U , U.

3.6.4. . p:XR l (p). , , X := X в R, X+ := epi p Tl : (x, t) t - l(x) (x X, t R). l (p) , Tl . : T X # , T [0, Tl ]. t1 := T (0, 1), l1 (x) := T (-x, 0); t2 := (Tl - T )(0, 1), l2 (x) := (Tl - T )(-x, 0).

, t1 0, t2 0, t1 + t2 = 1; l1 (t1 p), l2 (t2 p) l1 + l2 = l. t1 = 0, l1 = 0, . . T = 0 T [0, 1]Tl . t2 = 0, t1 = 1, . . T = Tl T [0, 1]Tl . t1 , t2 > 0. 1/t1 l1 (p) 1/t2 l2 (p), l = t1 (1/t1 l1 )+t2 (1/t2 l2 ). l ext( (p)), 3.6.2 l1 = t1 l, . . T = t1 T l . : l = 1 l1 +2 l2 , l1 , l2 (p) 1 , 2 > 0, 1 + 2 = 1. T := 1 Tl1 T := 2 Tl2 , T [0, Tl ], T + T = Tl . , [0, 1], T = Tl . (0, 1), 1 = . , l1 = l. l2 = l. 3.6.5. . p : X R . x X l ext( (p)) , l(x) = p(x). , . . , p : ext( (p)) = . X := X в R X+ := epi p X0 := 0 в R. , X+ X0 = 0 в R+ = epi 0. 3.6.4 X := 0, l := 0 p := 0, , T0 X0 . X0 X (. 3.5.3). 3.3.8, T X # T0 . ,

44

. 3.

T = Tl , l(x) := T (-x, 0) x X . 3.6.4, l ext( (p)). . 3.4.12 l ext(x (p (x))). 3.5.2 (2) 3.5.2 (4) : l ext(x (p)). 3.6.3 (1), x (p) (p). , 3.6.3 (2) l (p). 3.6.6. . p1 , p2 : X R . p1 p2 ( RX ) , (p1 ) ext( (p2 )). , p1 p2 (p1 ) (p2 ). , 3.6.5, p2 (x) = sup{l(x) : l ext( (p2 ))}. 3.7.

3.7.1. . (X, F, +, · ) F. (X, R, +, · |R вX ) (X, F, +, · ) XR . 3.7.2. . X f X# . Re f : x Re f (x) (x X ). Re : (X # )R (XR )# . 3.7.3. Re (X # )R (XR )# . F := C, F := R Re . Re . , Re (. 2.3.2). Re f = 0, 0 = Re f (ix) = Re(if (x)) = = Re(i(Re f (x) + i Im f (x))) = - Im f (x). f = 0 Re .

3.7.

45

g (XR )# , f (x) := g (x) - ig (ix). , f L (XR , CR ) Re f (x) = g (x) x X . , f (ix) = if (x), f X # . f (ix) = g (ix) + ig (x) = i(g (x) - ig (ix)) = if (x) , Re . 3.7.4. . Re . Re
-1 -1

: (XR )# (X # )R -

3.7.5. . 3.7.3 g : x g (x) - ig (ix) (g (XR )# , x X ).
-1

F := R Re .

-

3.7.6. . (X, F, +, · ) F. p : X R· , dom p = x1 , x2 X 1 , 2 F p(1 x1 + 2 x2 ) |1 |p(x1 ) + |2 |p(x2 ). 3.7.7. . ( ). 3.7.8. . p : X R· . | |(p) := {l X
#

: |l(x)| p(x) x X }

p. 3.7.9. . p : X R· | |(p) (p) | |(p) = Re
-1

( (p));

Re (| |(p)) = (p).

F := R | |(p) = (p). , Re . F := C. l | |(p), (Re l)(x) = Re l(x) |l(x)| p(x) x X , . . Re (| |(p)) (p). g (p) f := Re-1 g . f (x) = 0, |f (x)| p(x). f (x) = 0, := |f (x)|/f (x). |f (x)| = f (x) = f (x) = Re f (x) = g (x) p(x) = ||p(x) = p(x), || = 1. , f | |(p).

46

. 3.

3.7.10. X , p : X R X0 X . () | |(p + (X0 )) = | |(p) + | |( (X0 )). 3.7.9 3.5.5, : | |(p + (X0 )) = Re-1 ( (p + (X0 ))) = Re-1 ( (p) + ( (X0 ))) = = Re-1 ( (p)) + Re-1 ( ( (X0 ))) = | |(p) + | |( (X0 )). 3.7.11. X , Y , T L (X, Y ) p : Y R . p T , | |(p T ) = | |(p) T . 2.3.8 3.7.10, | |(p T ) = | |(p + (im T )) T = (| |(p) + | |( (im T ))) T = = | |(p) T + | |( (im T )) T = | |(p) T . 3.7.12. . 3.7.11 . 3.7.13. . X , p : X R X0 X . , , l0 X0 , |l0 (x0 )| p(x0 ) x0 X0 . l X , |l(x)| p(x) x X , , l(x0 ) = l0 (x0 ), x0 X0 . 3.8. 3.8.1. . R (. . R· -). X f : X R , t R {f t} := {x X : f (x) t};

3.8. {f = t} := f
-1

47

(t);

{f < t} := {f t} \ {f = t}. {f t}, {f = t}, {f < t} f . , {f = t} . 3.8.2. . T R t Ut (t T ) X . f : X R , {f < t} Ut {f t} (t T ) , t Ut . : T s t ( ). s < t, Us {f s} {f < t} Ut . : f (x) := inf {t f : X R. {f < t} , {f < t} Ut . s x Us s < t. , {f < t} f f (x) T : x Ut }. t T x {f < t}, f (x) < +, T , Us Ut . , x Ut , t, . . Ut {f t}.

3.8.3. , . f , g : X R (Ut )tT (Vt )tT : {f < t} Ut {f t}; {g < t} Vt {g t} (t T ). , , T R (. . ( r, t R, r < t) ( s T ) (r < X s < t)). f g ( R , . . f (x) g (x) x X ) , t1 , t2 T , t1 < t2 V
t1

Ut2 .

48 : V
t1

. 3.

{g t1 } {f t1 } {f < t2 } Ut2 .

: g (x) = + ( f (x) g (x)). t R , g (x) < t < +, t1 , t2 T g (x) < t1 < t2 < t. x {g < t1 } V
t1

Ut2 {f t2 } {f < t}.

, f (x) < t. - t : f (x) g (x). 3.8.4. . T R t Ut (t T ) . , , f : X R, {f < t} Ut {f t} (t T ). f {f < t} = {Us : s < t, s T }; {f t} = {Ur : t < r, r T } (t R). f 3.8.2 3.8.3. s < t, s T , Us {f s} {f < t}. f (x) < t, T s T , f (x) < s < t. , x {f < s} Us , {f < t}. r > t, r T . {f t} {f < r} Ur . , x Ur r T , r > t, f (x) r r > t, f (x) t. 3.8.5. X S . t R Ut := , t < 0, Ut := tS t 0. t Ut (t R) . 0 t x t2 S .
1

< t2 x t1 S , x (t1 /t2 ) t2 S . ,

3.8. 3.8.6. . pS : X R , {pS < t} tS {pS t} (t R+ )

49

{p < 0} = , S . ( 3.8.2, 3.8.4 3.8.5.) , pS (x) = inf {t > 0 : x tS } (x X ). 3.8.7. . . , , p , {p < 1} {p 1} . p S , {p < 1} S {p 1}. S pS . x X . pS (x) 0 . > 0. pS (x) = inf {t > 0 : x tS } = inf t > 0 : x = inf { > 0 : x S, > 0} = = inf { > 0 : x S } = pS (x). pS x1 , x2 X , , t1 , t2 > 0 t1 S + t2 S (t1 + t2 )S ( t1 x1 + t2 x2 = (t1 + t2 ) pS (x1 + x2 ) = inf {t > 0 : x1 + x2 tS } inf {t : t = t1 + t2 ; t1 , t2 > 0, x1 t1 S, x2 t2 S } = = inf {t1 > 0 : x1 t1 S } + inf {t2 > 0 : x2 t2 S } = pS (x1 ) + pS (x2 ). t2 t1 x1 + x2 t1 + t2 t1 + t 2 , t S =

50

. 3.

p : X R· . {p < 1} S {p 1}. Vt := {p < t}, Ut := tS t R+ Vt := Ut := t < 0. , {pS < t} Ut {pS t}; {p < t} Vt {p t}

t R. 0 t1 < t2 , Vt1 = {p < t1 } = t1 {p < 1} t1 S = Ut1 Ut2 . , Ut1 t1 {p 1} {p t1 } {p < t2 } Vt2 . , 3.8.3 3.8.4, p = pS . 3.8.8. . S X , dom pS = X . , S , pS . p {p < 1} {p 1} . 3.8.9. . H X , X/H . X/H X ( H ). X , - X . XR X X . 3.8.10. X X # . 3.8.11. . X , U X L X . L U = , H X , H L H core U = . , , core U = ( ) , , 0 core U . x L X0 := L - x. - X/X0 : X X/X0 . 3.1.8 3.4.10, , (U ) . , 3.8.7 3.8.8 p := p(U ) , dom p = X/X0 , , (core U ) core (U ) {p < 1} (U ).

51

, , , p((x)) 1 (x) (U ). 3.5.6 f x (p ). 3.5.3, f x (p ) =
(x)

(p) .

H := {f = p (x)}. , H X . , H L, . 3.5.2 (1), : H core U = . f := Re-1 f H := {f = f (x)}. , L H H . , H . 3.8.12. . 3.8.11 , core U L = . , 3.8.11 . 3.8.13. . U , V X H X . , H U V , , H , . . H = {f = t}, f (XR )# t R, V {f t} U {f t} := {-f -t}. 3.8.14. U . U V
3.1. , , . 3.2. , . 3.3. , , . . . . 3.4. . , ?

. U V , V , V .

52
3.5. S1 S2 S = 0 S S1 S2 .

. 3.
1

S1 (1 - )S2 . ,

3.6. , , . 3.7. S := {p + q 1}, p, q Sp Sq . S Sp Sq . 3.8. , RN . 3.9. . 3.10. p, q , , . . , dom p - dom q = dom q - dom p. (. 3.5.7)

(p + q ) = p + q . 3.11. p, q : X R X . (p q ) = co( p q ). 3.12. . 3.13. 2 в 2- , . ? ? 3.14. ? 3.15. R
N

K -?

3.16. ? 3.17. l .

-

3.18. , .

3.19. C X Ё H (C ) H (C ) = {(x, t) X в R : x tC }. Ё.

4

4.1. 4.1.1. . d : X 2 R+ X , (1) d(x, y ) = 0 x = y ; (2) d(x, y ) = d(y , x) (x, y X ); (3) d(x, y ) d(x, z ) + d(z , y ) (x, y , z X ). (X, d) . d(x, y ) , X .

x y.

4.1.2. d : X 2 R+ , (1) {d 0} = IX ; (2) {d t} = {d t}-1 (t R+ ); (3) {d t1 } {d t2 } {d t1 + t2 } (t1 , t2 R+ ). 4.1.2 (1)4.1.2 (3) 4.1.1 (1) 4.1.1 (3) . 4.1.3. . (X, d) R+ \ 0. B := Bd, := {d }

( ), B := B

d,

:= {d < }

54

. 4.

( ). B (x) x B x.

B (x) x. 4.1.4. , ґ , . 4.1.5. . , (X, d) X 2 , UX , Ud , , , U , , . X := UX := {}. UX (). 4.1.6. U .
-1

(1) U fil {IX }; (2) U U U U; (3) ( U U ) ( V U ) V V U ; (4) {U : U U } = IX . 4.1.7. . 4.1.6 (4), 4.1.1 (1), U . 4.1.8. X UX (x) := {U (x) : U U }. (x) x X . (1) (x) fil {x}; (2) ( U (x)) ( V (x) & V U ) ( y V ) V (y ).

4.1.9. . : x (x) , (x) x. : X , (U ) . .

4.1.

55

4.1.10. . . . , X . , 4.1.6 (4), X . 4.1.11. . G X , (: G Op( ) (( x G) G (x))). F X , (: F Cl( ) (X \ F Op( ))). 4.1.12. . . 4.1.13. . U X int U := U := {G Op(X ) : G U }; cl U := U := {F Cl(X ) : F U }. int U U , U . cl U U , U . X \ U U , U . X , , U , U . U U fr U U . 4.1.14. U x , x U . 4.1.15. . 4.1.14 Op(X ) X , , X Op(X ). , , Cl(X ) X . 4.1.16. . B X . , B x X x B ( : B x), fil B x, . . fil B (x).

56

. 4.

4.1.17. . (x ) () X . , x (: x x), x . . , x = lim x x (x ), . 4.1.18. . , , . . 4.1.19. U x : (1) x U ; (2) F , F x U F ; (3) (x ) U , x. (1) (2): x U , (x) fil {U } F := (x) fil {U }. (2) (3): F x U F . F , . xV V U V F . , xV x. (3) (1): V , (x ) V x x. , x V . , x X \ V x X \ V . 4.1.20. . 4.1.19 (2) , F , 4.1.19 (3) := N. : . 4.2. 4.2.1. f : X Y X , Y . : X Y -

4.2. (1) (2) (3) (4) (5)

57

G Op(Y ) f -1 (G) Op(X ); F Cl(Y ) f -1 (F ) Cl(X ); f (X (x)) Y (f (x)) x X ; (x X, F x) (f (F ) f (x)) F ; f (x ) f (x), x (x ). (1) (2) 4.1.11. , (1) (3) (4) (5) (2). (1) (3): V Y (f (x)), W := int V Op(Y ) f (x) W . f -1 (W ) Op(X ) x f -1 (W ). , f -1 (W ) X (x) (. 4.1.14). , f -1 (V ) f -1 (W ) , , f -1 (V ) X (x). , V f (f -1 (V )). (3) (4): F x, fil F X (x) 4.1.16. , f (F ) f (X (x)) Y (f (x)). 4.1.16 f (F ) f (x). (4) (5): (x ) f (f (x )) . (5) (2): F Y . F = , f -1 (F ) , . F x f -1 (F ). (x ) f -1 (F ), x ( 4.1.18). f (x ) F f (x ) f (x). 4.1.18, , f (x) F , , x f -1 (F ). 4.2.2. . f : X Y , ( , ) 4.2.1 (1)4.2.1 (5), ( ) . 4.2.1 (5) x X , , f x. , f X , f X . 4.2.3. . 4.2.1 (5). 4.2.4. f : X Y UX , UY X Y . : (1) ( V UY ) ( U UX ) ( x, y )(x, y ) U (f (x), f (y )) V ;

58

. 4. f UX ; f в : X 2 Y (x), f (y )); ) UX , . . f в-1 (UY ) UX . 1.1.10 U X 2 V Y 2

(2) ( V UY ) f -1 V (3) f в (UX ) UY , f в : (x, y ) (f (4) ( V UY ) f в-1 (V , f
-1

V f =
(v1 ,v2 )V

f

-1

(v1 ) в f

-1

(v2 ) =
в-1

= {(x, y ) X 2 : (f (x), f (y )) V } = f f U f
-1

(V );

=
(u1 ,u2 )U

f (u1 ) в f (u2 ) =

= {(f (u1 ), f (u2 )) : (u1 , u2 ) U } = f в (U ). 4.2.5. . f : X Y , ( , ) 4.2.4 (1)4.2.4 (4), ( ) . 4.2.6. . f : X Y , g : Y Z h := g f : X Z . , hв (x, y ) = (h(x), h(y )) = (g (f (x)), g (f (y ))) = = g в (f (x), f (y )) = g в f в (x, y ) x, y X . , hв (UX ) = g в (f в (UX )) g в (UY ) UZ 4.2.4 (3). 4.2.4 (3), , h . 4.2.7. . 4.2.8. . E X Y UX , UY . E ( ) , ( V UY )
f E

f

-1

V f UX .

4.3.

59

4.2.9. . . 4.3. 4.3.1. (X1 , d1 ) (X2 , d2 ) . , , X := X1 в X2 . x := (x1 , x2 ) y := (y1 , y2 ) d(x, y ) := d1 (x1 , y1 ) + d2 (x2 , y2 ). d X . x := (x1 , x2 ) X X (x) = fil{U1 в U2 : U1
X1

(x1 ), U2

X2

(x2 )}.

4.3.2. . X X1 X2 X1 X2 X1 в X2 . 4.3.3. . f : X R· , epi f X R.

4.3.4. . (1) f : X R . (2) f : X R· , f (x) := sup{f (x) : } (x X ) , epi f = epi f . 4.3.5. f : X R· , x X f (x) = lim inf f (y ).
y x

, ,
y x

lim inf f (y ) := lim f (y ) := sup inf f (U )
y x U (x)

60

. 4.

f x ( (x)). : x dom f , (x, t) epi f t R. , Ut x, inf f (Ut ) > t. : limyx inf f (y ) = + = f (x). x dom f , inf f (V ) > - V x. > 0 U (x), V , xU U inf f (U ) f (xU ) - . xU dom f , , xU x ( x). tU := inf f (U ) + . , tU t := limyx inf f (y ) + . (xU , tU ) epi f , (x, t) epi f f .
y x

lim inf f (y ) + f (x) lim inf f (y ).
y x

: (x, t) epi f , t < lim inf f (y ) = sup inf f (U ).
y x U (x)

, inf f (U ) > t U x. , (X в R) \ epi f . 4.3.6. . , 4.3.5, . 4.3.7. f : X R , f -f . 4.3.8. f : X R· , t R {f t}. : x {f t}, t < f (x). 4.3.5 U x t < inf f (U ). , X \ {f t} . : - x X t R limyx inf f (y ) t < f (x). > 0 t + < f (x) , 4.3.5, U (x) xU U {f inf f (U ) + }. , xU {f t + } xU x. .

4.4. 4.4.

61

4.4.1. . C X . C , E Op(X ) , C {G : G E }, E0 E , C {G : G E0 }. 4.4.2. . 4.4.1 : , . 4.4.3. . . 4.4.4. . 4.4.3 , . . , . 4.4.5. . . . 4.4.6. (. . ). , f : X R· X . t0 := inf f (X ). t0 = +, . t0 < +, T := {t R : t > t0 }. Ut := {f t} t T . , {Ut : t T } ( x : f (x) = inf f (X )). . {Gt := X \ Ut : t T } X . {Gt : t T0 }, : {Ut : t T0 } = . , Ut1 Ut2 = Ut1 t2 t1 , t2 T . 4.4.7. . , (. 9.4.4).

62

. 4. 4.4.8. . .

4.4.9. . . 4.5. 4.5.1. B X . {B 2 : B B } в 2 B X . (B1 в B1 ) (B2 в B2 ) (B1 B2 ) в (B1 B2 ) 4.5.2. . F X UX X . F , F в UX . X , . . 4.5.3. . V X 2 , U X , , U V , U 2 V . , U B , diam U := sup(U 2 ) . : , . 4.5.4. : (1) ; (2) ; (3) . (1) (2) (3) , (3) (1). Un F , B1/n . Vn := U1 . . . Un xn Vn . , V1 V2 . . . diam Vn 1/n. , (xn ) . , : x := lim xn . , F x. n0 N : d(xm , x) 1/2n m n0 . n N d(xp , y ) diam Vp 1/2n

4.5.

63

d(xp , x) 1/2n, p := n0 2n y Vp . , y Vp d(x, y ) 1/n, . . Vp B1/n (x). : F (x). 4.5.5. . , ( ) 4.5.4 (1)4.5.4 (3), ( ) . 4.5.6. . , , , . : B , , 1.3.1, B . B , . . : B x. x . : F . B := {cl V : V F }. B . , x , x cl V V F . , F x. , V F /2 y V . y V d(x, y ) /2 , , d(x, y ) d(x, y ) + d(y , y ) , . ., , V B (x) , , B (x) F . 4.5.7. , B1 (x1 ) . . . Bn (xn ) Bn+1 (xn+1 ) . . . , (n ) , . 4.5.8. . f X UX Y UY . , , F X . V UY , f -1 V f UX 4.2.5 (. 4.2.4 (2)). F , U F U 2 f -1 V f . , f (U ) V . , f (U )2 =
(u1 ,u2 )U
2

f (u1 ) в f (u2 ) =
-1

= f U2 f

-1

f (f

-1

V f) f
-1

= (f f

-1

) V (f f

-1

) V,

, 1.1.6, f f

= Iim f IY .

64

. 4. 4.5.9. 4.5.8 4.5.4. .

4.5.10. X0 X (. . cl X0 = X ) f0 : X0 Y X0 Y . , , f : X Y , f0 , . . , f |X0 = f0 . x X Fx := {U X0 : U X (x)} X0 . , 4.5.8 , f0 (FX ) Y . Y y Y , . . f0 (Fx ) y . , (. 4.1.18). f (x) := y . f . 4.5.11. . f : (X, d) (X , d ) X X ( ), d = d f в . f X X (, ), f X X , , im f = X . 4.5.12. . (X, d) . (X , d ) : (X, d) (X , d ) (X , d ). (X , d ) , (X, d) - (X , d ) @ 1@ R @ ? (X1 , d1 ) 1 : (X, d) (X1 , d1 ) X (X1 , d1 ), : (X , d ) (X1 , d1 ) X X1 . 4.5.10. , 0 := 1 -1 . 0 (X ) X 1 (X ) X1 .

4.6.

65

0 X . , X1 . x1 X1 . (1 (xn )), xn X . , (xn ) . , ((xn )) X . x := lim (xn ), x X . (x) = lim 0 ((xn )) = lim 1 -1 ((xn )) = lim 1 (xn ) = x1 . X . X X . X : x1 x2 d(x1 (n), x2 (n)) 0. X := X / d((x1 ), (x2 )) := lim d(x1 (n), x2 (n)), : X X . : (X, d) (X , d ) : (x) := (n x (n N)). 4.5.13. . (X , d ), 4.5.12, ґ , (X, d). 4.5.14. . X0 (X, d) 2 , (X0 , d|X0 ) (X, d). 4.5.15. . . 4.5.16. X0 X . X0 X0 X . X := cl X0 : X0 X . , . X 4.5.15. 4.5.12. 4.6. 4.6.1. . 4.6.2. . U X V UX . E X V - U , U V (E ). 4.6.3. . , V UX V -.

66

. 4.

4.6.4. V UX U X V -, U . V UX W UX , W W V . W - F U , . . U W (F ). F , W - E F , . . F W (E ). U W (F ) W (W (E )) = W W (E ) V (E ), . . E V - U . U , U1 , . . . , Un U1 , . . . X V UX U , U = U1 . . . Un , Un V .

4.6.5.

4.6.6. . , 4.6.5, : , . 4.6.7. . , . 4.6.8. C (X, F) X F Ё d(f , g ) := sup dF (f (x), g (x)) = sup |f (x) - g (x)| (f , g C (X, F)).
xX xX

UF U := (f , g ) C (X, F)2 : g f Ud = fil {U : UF }. 4.6.9. C (X, F) . 4.6.10. . E C (X, F) , E {g (X ) : g E } F.
-1

.

4.6.

67

: , {g (X ) : g E } , . E UF . U - E E . U UX , U := f -1 f
f E

(. 4.2.9). g E f E , g f =
-1

-1

,

(g f

-1 -1

)

= (f

-1 -1

)

g

-1

=f g

-1

.

, 4.6.8 g в (U ) = g U g -1 g (f -1 f ) g -1 (g f
-1

) (f g

-1

) .

g , E . : 4.5.15, 4.6.7, 4.6.8 4.6.9 UF U - E . UF , , U UX , U
g E

g

-1

g

( U E ). , {U (x) : x X } X . X , {U (x0 ) : x0 X0 }. , 1.1.10 IX
x0 X0 -1

U (x0 ) в U (x0 ) =

=
(x0 ,x0 )I
X0

U

(x0 ) в U (x0 ) = U IX0 U.

68

. 4.

{g |X0 : g E } FX0 . , -. , E E , , g E f E g IX0 f
-1

.

, gf
-1

= g IX f
-1

-1

g (U IX0 U ) f
-1

-1

g (g = (g g

g ) IX0 (f

f) f

-1 -1

= )=

-1

) (g IX0 f

-1

) (f f

= Iim g (g IX0 f

-1

) Iim f

. , 4.6.8, E U - E .

4.6.11. . - . : , U , /3 , U . ( ) , . 4.7. 4.7.1. . U , , . . int cl U = . U ( ), U ( ) , . . U nN Un , int cl Un = . , . . , , . 4.7.2. . , .

4.7.

69

4.7.3. : (1) X ; (2) ; (3) (. . X ) ; (4) . (1) (2): U := nN Un , Un = cl Un , int Un = . U . int U U int U , int U , , X . (2) (3): U := nN Gn , Gn cl Gn = X . X \U = X \nN Gn = nN (X \Gn ). X \Gn int(X \ Gn ) = ( cl Gn = X ). , int(X \ U ) = . , U , . . U . (3) (4): U X , . . U nN Un int cl Un = . , Un = cl Un . Gn := X \ Un . nN Gn = X \ nN Un . X \ U , , X \ U . (4) (1): U X , X \ U . , U . 4.7.4. . 4.7.3 (4) , () . . 4.7.5. . X (f : X R) , sup{f (x) : } < + x X . G X G0 , (f ) , . . supxG0 sup {f (x) : } +. 4.7.6. . . G x0 G. , G , . . G nN Un , int Un = Un = cl Un .

70

. 4.

0 > 0 B0 (x0 ) G. , U1 B0 /2 (x0 ), . . x1 B0 /2 (x0 ) \ U1 . U1 1 , 0 < 1 0 /2 B1 (x1 ) U1 = . , B1 (x1 ) B0 (x0 ). , d(x1 , y1 ) 1 , d(y1 , x0 ) d(y1 , x1 ) + d(x1 , x0 ) 1 + 0 /2, d(x1 , x0 ) 0 /2. B1 /2 (x1 ) U2 . x2 B1 /2 (x1 ) \ U2 0 < 2 1 /2 , B2 (x2 ) U2 = . , B2 (x2 ) B1 (x1 ). , B0 (x0 ) B1 (x1 ) B2 (x2 ) . . . , n+1 n /2 Bn (xn ) Un = . 4.5.6 x := lim xn . , , x = nN Un , , x G. , x B0 (x0 ) G. . 4.7.7. . . . f : [0, 1] R x [0, 1) D+ f (x) := lim inf
h0

f (x + h) - f (x) ; h f (x + h) - f (x) . h

D+ f (x) := lim sup
h0

D+ f (x) D+ f (x) R f x. D f C ([0, 1], R), x [0, 1) D+ f (x) D+ f (x) R, . . . D . , , (0, 1), C ([0, 1], R). . :

n=0

4n x 4n

4.8. ( x := (x - [x]) (1 + [x] - x) x ),

71

+

n=0

1 sin (n2 x) n2

, ,

n=0

bn cos (an x)
3 2

( a

, 0 < b < 1 ab > 1 +

).

4.8. 4.8.1. . , , , , R2 . , .

4.8.2. . (= ) ( ) . () . . . 4.8.3. . R2 . G1 G2 , G1 G2 = R2 \ ; = G 1 = G 2 . 4.8.4. . G1 G2 , 4.8.3, . , , . . . : .

72

. 4.

4.8.5. . D, D1 , . . . , Dn (= ) , Dm Dk = m = k D1 , . . . , Dn int D.
n

D\
k=1

int Dk

. , (= ) , . . 4.8.6. . F F . F () R2 F () () F . , 4.8.3 , , . 4.8.7. K G , K . F , K int F F G. 4.8.8. . F , 4.8.7, (K, G).
4.1. . , . 4.2. X 2 , X ? 4.3. S [0, 1]
1

d(f , g ) :=
0

|f (t) - g (t)| dt 1 + |f (t) - g (t)|

(f , g S )

( .

?). -

4.4. , NN d(, ) = 1/ min {k N : k = k }. , d N .
N

73

4.5. ? ? 4.6. ? ? 4.7. , . 4.8. A B R d(A, B ) :=
xA y B N

-

sup inf |x - y |

y B xA

sup inf |x - y |

.

, d . . ? 4.9. , RN . ? 4.10. , R .
N

4.11. ( ) . 4.12. , . 4.13. . 4.14. (Y , d) . F : Y Y , d(F (x), F (y )) d(x, y ) > 1 x, y Y . F : Y Y Y . , F . 4.15. , . 4.16. . 4.17. ? 4.18. ? 4.19. ? ?

5

5.1. 5.1.1. F p : X R· (1) (2) (3) X . dom p X ; p(x) 0 x X ; ker p := {p = 0} X;

(4) B p := {p < 1} Bp := {p 1} , p B , B p B Bp ;

(5) X = dom p , B . x1 , x2 dom p 1 , 2 F, 3.7.6 p(1 x1 + 2 x2 ) |1 |p(x1 ) + |2 |p(x2 ) < + + (+) = +.

p

, (1) . , (2) , . . x X p(x) < 0. 0 p(x) + p(-x) < p(-x) = p(x) < 0. . (3) (2) p. (4) (5) (. 3.8.8). .

5.1.

75

5.1.2. p, q : X R· . p q ( (R· )X ) , Bp Bq . : , {q 1} {p 1}. : , 5.1.1 (4), p = pBp q = pBq . t1 , t2 R , t1 < t2 . t1 < 0, {q t1 } = , , {q t1 } {p t2 }. t1 0, t1 Bq t1 Bp t2 Bp . , 3.8.3, p q . 5.1.3. X , Y pT (x) := inf p T (x) BT := T -1 (Bp x1 , x2 X , T X в Y p : Y R· . , , x X . pT : X R· , ) , pT = pBT . 1 , 2 F

pT (1 x1 + 2 x2 ) = inf p(T (1 x1 + 2 x2 )) inf p(1 T (x1 ) + 2 T (x2 )) inf (|1 |p(T (x1 )) + |2 |p(T (x2 ))) = = |1 |pT (x1 ) + |2 |pT (x2 ), . . pT . , BT , 5.1.1 (4) 3.1.8. x BT , y Bp (x, y ) T . pT (x) p(y ) 1, . . BT BpT . , , xB
pT

, pT (x) = inf {p(y ) : (x, y ) T } < 1. , y
-1

T (x) , p(y ) < 1. , x T , B
pT

(B p ) T

-1

(Bp ) = BT . = pT .

BT B

pT

. 5.1.1 (4), : p

B

T

5.1.4. . pT , 5.1.3, p T . 5.1.5. . p : X R ( 3.4.3 , dom p = X ). (X, p) . , , X . 5.1.6. . ( RX ) MX

76

. 5.

M, , X . (X, MX ), ґ X , . 5.1.7. M (R· )X , (X, p) p M. 5.1.8. . MX ( ), x X, x = 0, p MX , p(x) = 0. X ( ) . 5.1.9. . , , . X ( ) X · () · X , · | X , X . (X, · ) . , X . 5.1.10. . (1) (X, p) (X, {p}). . (2) M ( ) X . M , X . (3) (Y , N) T XвY , dom T = X . 3.4.10 5.1.1 (5) p N pT , , M := {pT : p N} X . N N T () NT . , T L (X, Y ), M = {p T : p N}. N T := M. , X Y0 Y T T := : Y0 Y . Y0 , , N . ,

5.1.

77

N Y0 . . (4) F , , | · | : F R. X f X # . f : X F, : pf (x) := |f (x)| (x X ). X X # , (X, X ) := {pf : f X } X , X . (5) (X, p) , X0 X : X X/X0 . -1 X/X0 . , p-1 , p X0 pX/X0 . (X/X0 , pX/X0 ) - (X, p) X0 . - 5.3.11. (6) X M (R· )X . M X0 := {dom p : p M}. , (X0 , {p : p M}), X0 X , : M () , M . : p
,

(f ) := sup |x f (x)| : ,
xR
N

() RN ( , . 10.11.6). (7) (X, · ) (Y , · ) ( F). T L (X, Y ) , . . T := sup { T x : x X, x 1} = sup
xX

Tx . x

78

. 5.

( , 0/0 := 0.) , · : L (X, Y ) R· . , BX := { · X 1}, T1 , T2 L (X, Y ) 1 , 2 F 1 T1 + 2 T2 = sup · 1 T1 +2 T2 (BX ) = = sup · ((1 T1 + 2 T2 )(BX )) sup 1 T1 (BX ) + 2 T2 (BX ) |1 | sup ·
T1

(BX ) + |2 | sup ·
1

T2

(BX ) =

= |1 | T

+ |2 | T2 .

B (X, Y ), , , . , B (X, Y ) ( ). , T L (X, Y ) , , . . K , T x Y K x X (x X ). T K , . (8) X F · X . , , X := B (X, F) , . . f : f = sup{|f (x)| : x 1} = sup
xX

|f (x)| . x

X := (X ) := B (X , F) X . x X f X x := (x) : f f (x). , (x) (X )# = L (X , F). , x = (x) = sup {|(x)(f )| : f
X

1} =

5.2. = sup{|f (x)| : |f (x)| x = sup{|f (x)| : f | |( ·
X X

79 (x X )} = )} = x
X

.

, , 3.6.5 3.7.9. , (x) X x X . , : X X , : x (x), , x = x x X . X , , . , , x x := x , . . X X . X , X X ( ). . , , . , , C ([0, 1], F). 5.1.11. . , 5.1.10 (8), ( ) X X . x X f X ( f x) (x, f ) := x | f := f (x). X x , . . x | x = (x, x ) = x (x). 5.2. 5.2.1. (X, p) . x1 , x2 X dp (x1 , x2 ) := p(x1 - x2 ). (1) dp (X 2 ) R+ , {d 0} IX ; (2) {dp t} = {dp t}-1 , {dp t} = t{dp 1} (t R+ \ 0); (3) {dp t1 } {dp t2 } {dp t1 + t2 } (t1 , t2 R+ ); (4) {dp t1 } {dp t2 } {dp t1 t2 } (t1 , t2 R+ ); (5) p dp . 5.2.2. . Up := fil {{dp t} : t R+ \ 0} (X, p).

80

. 5. Up -

5.2.3. . (1) (2) (3)

Up fil {IX }; U Up U -1 Up ; ( U Up ) ( V Up ) V V U . M) p M} UM , UX 5.2.3 (1) 1.3.13.)

5.2.4. . (X, . U := sup{Up : X ( . .). ( U

5.2.5. (X, M) . (1) U fil {IX }; (2) U U U
-1

U;

(3) ( U U ) ( V U ) V V U . (3). U U , 1.2.18 1.3.8 p1 , . . . , pn M , U = U{p1 ,...,pn } = Up1 . . . Upn . 1.3.13, Uk Upk U U1 . . . Un . 5.2.3 (3), Vk Upk , Vk Vk Uk . , (V1 . . . Vn ) (V1 . . . Vn ) V1 V1 . . . Vn Vn U1 . . . Un . , V1 . . . Vn Up1 . . . Upn U . 5.2.6. M X , UM , . . {V : V UM } = IX . : (x, y ) IX , . . x = y . p M p(x - y ) > 0. , (x, y ) {dp 1/2 p(x - y )}. Up , UM . , X 2 \ IX X 2 \ {V : V UM }. , IX {V : V UM }. : p(x) = 0 p M. (x, 0) V V UM , , (x, 0) IX . , x = 0.

5.2. 5.2.7. X UX (x) := {U (x) : U UX } (x X ). (x) (1) (x) (2) ( U (.

81

x X . fil {x}; (x)) ( V (x) & V U ) ( y V ) V (y ). 4.1.8).

5.2.8. . : x (x) (X, M), (x) x. : X , M , (UM ) . . 5.2.9. x X X (x) = sup{p (x) : p MX }. 5.2.10. X . x X U (x) U - x X (0). 5.2.9 1.3.13 (X, p). > 0 {dp }(x) = Bp + x, Bp := {p 1}. , p(y -x) , y = (-1 (y -x))+x -1 (y -x) Bp . , y Bp + x, p(y - x) = inf {t > 0 : y - x tBp } . 5.2.11. . 5.2.10 , () (X, p). X Bp , BX . . 5.2.12. MX , X , . . x1 , x2 X U1 X (x1 ) U2 X (x2 ) , U1 U2 = .

82

. 5.

: x1 = x2 p MX := p(x1 - x2 ) > 0. U1 := x1 + /3 Bp , U2 := x2 + /3 Bp . 5.2.10, Uk X (xk ). , U1 U2 = . , y U1 U2 , p(x1 - y ) /3 p(x2 - y ) /3. p(x1 - x2 ) 2/3 < = p(x1 - x2 ), . : (x1 , x2 ) {V : V UX }, x2 {V (x1 ) : V UX }. x1 = x2 , , 5.2.6 MX . 5.2.13. . , , , , , , . . 5.2.14. (X, p) X0 X . - (X/X0 , pX/X0 ) , X0 . : x X0 . (x) = 0, , , : X X/X0 . 0 = := pX/X0 ((x)) = p-1 ((x)) = inf {p(x + x0 ) : x0 X0 }. , x + /2 Bp X0 , . . x X0 . , X0 . - X/X0 : x x = (x) x X . pX/X0 (x) = 0, 0 = inf {p(x - x0 ) : x0 X0 }, . . (xn ) X0 , xn x. , 4.1.19, x X0 x = 0. . 5.2.15. - -. U ( ) U = ( ). 4.1.9 x, y cl U (x ), (y ) U , x x, y y . (, ) , x + y U . 4.1.19, x + y = lim(x + y ) cl U . 5.3. 5.3.1. . M N . , M N, M N, UM UN . M NN M, , M N , M N.

5.3.

83

5.3.2. . M N X : (1) M N; (2) x X M (x) N (x); (3) M (0) N (0); (4) ( q N) ( p1 , . . . , pn M) ( 1 , . . . , n R+ \ 0) Bq 1 Bp1 . . . n Bpn ; (5) ( q N) ( p1 , . . . , pn M) ( t > 0) q t(p1 . . . pn ) ( K - RX ). (1) (2) (3) (4): . (4) (5): (. 5.1.2), q pBp1 /1 . . . pBpn /n = = 1 p 1
1

...

1 p n

n

1 1 ... 1 n

p1 . . . pn .

(5) (1): , M {q } q N. V Uq , V {dq } > 0. . . . dpn {dq } dp1 t t p1 , . . . , pn M t > 0. , , Up1 . . . Upn = U{p1 ,...,pn } UM . , V UM . 5.3.3. . p, q : X R X . , p q , p q , {p} {q }. p q . 5.3.4. p q ( t > 0) q tp ( t 0) Bq tBp ;

p q ( t1 , t2 > 0) t2 p q t1 p ( t1 , t2 > 0) t1 Bp Bq t2 Bp . 5.3.2 5.1.2. 5.3.5. . p, q : FN R FN . p q ker p ker q . 5.3.6. . .

84

. 5.

5.3.7. (X, M) (Y , N) T L (X, Y ) . : (1) N T M; (2) T в (UX ) UY , T в-1 (UY ) UX ; (3) x X T (X (x)) Y (T x); (4) T (X (0)) Y (0), X (0) T -1 (Y (0)); (5) ( q N) ( p1 , . . . , pn M) q T p1 . . . pn . 5.3.8. (X, · X ) (Y , · Y ) T L (X, Y ) . : (1) T (. . T B (X, Y )); (2) · X · Y T; (3) T ; (4) T ; (5) T . 5.3.7. 5.3.9. . 5.3.7 , M - ( ) . M := {sup M0 : M0 M}. . 5.3.10. . X := F X0 X0 := F1, 1 : 1 ( ). M := {p : }, p (x) := |x( )| (x F ). , M X . : X X/X0 . , M-1 . M-1 . 5.3.11. . (X, M) X0 X . M-1 , : X X/X0 , MX/X0 . (X/X0 , MX/X0 )

5.4.

85

- X X0 . 5.3.12. - X/X0 , X0 . 5.4. 5.4.1. . (X, M) . (X, M) , d X , UM = Ud . X , M, X . X , , X . 5.4.2. . , . : UM = Ud . , , M, , n N pn M tn > 0, {d 1/n} {dpn tn }. N := {pn : n N}. , M N. V UM , V {d 1/n} n N . , V Upn UM , . . M N. , M N. Ud 4.1.7. 5.2.6, , UM UN . : , , , , : M := {pn : n N} M . x1 , x2 X

d(x1 , x2 ) :=
k=1

1 pk (x1 - x2 ) 2k 1 + pk (x1 - x2 )

( k k=1 1/2 , d ). , d . . , (t) := t(1 + t)-1 (t R+ ). , (t) = (1 + t)-2 > 0.

86

. 5.

, . : (t1 + t2 ) = (t1 + t2 )(1 + t1 + t2 )-1 = = t1 (1 + t1 + t2 )-1 + t2 (1 + t1 + t2 )-1 t1 (1 + t1 )-1 + t2 (1 + t2 )-1 = = (t1 ) + (t2 ). , x, y , z X

d(x, y ) =
k=1

1 (pk (x - y )) 2k

k=1

1 (pk (x - z ) + pk (z - y )) 2k

k=1

1 ((pk (x - z ) + (pk (z - y ))) = d(x, z ) + d(z , y ). 2k

Ud UM . , Ud UM . {d }, (x, y ) {dp1 t} . . . {dpn t}.
n

d(x, y ) =
k=1

1 pk (x - y ) + 2k 1 + pk (x - y )
n

k=n+1

1 pk (x - y ) 2k 1 + pk (x - y )

t 1+t

k=1 -n

1 + 2k

k=n+1

1 t 1 + . 2k 1 + t 2n t 0, } UM , {d }.

t(1 + t) + 2 , n t n (x, y ) {d }. , {d . , UM Ud . pn M t > 0 > 0 , {dpn t} , := 1t 1 pn (x - y ) d(x, y ) = n n 1 + p (x - y ) 2 2 1+t n x, y , pn (x - y ) t. 1t , 2n 1 + t

-1

5.5.

87

5.4.3. . V (X, M) , sup p(V ) < + p M, . . p(V ) R p M. 5.4.4. V (X, M) : (1) V ; (2) (xn )nN V (n )nN F , n 0, n xn 0 (. . p(n xn ) 0 p M); (3) V . (1) (2): p(n xn ) |n |p(xn ) |n | sup p(V ) 0. (2) (3): U X (0) , U V . 3.4.9 , ( n N) ( xn V ) xn nU . , 1/n xn U n N, . . (1/n xn ) . (3) (1): p M. n N, V nBp . , sup p(V ) sup p(nBp ) = n < +. 5.4.5. . , . : . : V . , , V = Bp p M. , p M. U M (0), nU V n N. , U p (0). 5.3.2, , p M. , p M , , 5.2.12, p . , p . 5.4.6. . 5.4.5 , . 5.5. 5.5.1. . .

88

. 5.

5.5.2. . . , , . 5.5.3. , (= ) . : n=1 xn < + (xn ). sn := x1 + . . . + xn , m > k
m m

sm - sk =
n=k+1

xn
n=k+1

xn 0.

: (xn ) . (nk )kN , xn - xm 2-k n, m nk . xn1 + (xn2 - xn1 ) + (xn3 - xn2 ) + . . . x, . . xnk x. , xn x. 5.5.4. X X0 X . - X/X0 . : X X := X/X0 . , x X x -1 (x) , 2 x x x . , n=1 xn , X , -1 xn (xn ), n=1 xn . 5.5.3 x := n=1 xn . x := (x).
n n

x-
k=1

xk x -
k=1

xk 0.

5.5.3, , X . 5.5.5. . , 5.5.3 . , (X, p) , - X/ ker p .

5.5.

89

5.5.6. . X , Y X = 0. B (X, Y ) , Y . : (Tn ) B (X, Y ). x X Tm x - Tk x Tm - Tk x 0, . . (Tn x) Y . , T x := lim Tn x. , T . | Tm - Tk | Tm - Tk ( Tn ) R, , . . supn Tn < +. , Tn x supn Tn x , : T < +. , Tn - T 0. > 0 n0 , Tm - Tn /2 m, n n0 . , x BX m n0 , Tm x - T x /2. Tn x - T x Tn x - Tm x + Tm x - T x Tn - Tm + Tm x - T x n n0 . , Tn - T = sup{ Tn x - T x : x BX } n. : (yn ) Y . x X x = 1. 3.5.6 3.5.2 (1), x | |( · ), (x, x ) = x = 1. , Tn := x yn : x (x, x )yn B (X, Y ), Tn = x yn . , Tm - Tk = x (ym - yk ) = x ym - yk = ym - yk , . . (Tn ) B (X, Y ). T := lim Tn . T x - Tn x = T x - yn T - Tn x 0. , T x (yn ) Y . 5.5.7. . ( ) . 5.5.8. . X , : XX , X X . cl (X ) X . 5.5.7, X . 5.1.10 (8) X X . 4.5.16. 5.5.9. . (1) : ,

90

. 5.

, , 5.5.45.5.8. (2) E . x F E x := sup |x(E )|. l (E ) := l (E , F) := dom · E . : B (E ) B (E , F). E := N m := l := l (E ). (3) F E . x c(E , F ) (x l (E ) x(F ) F). , E := N F N, c := c(E , F ) . c(E , F ) c0 (E , F ) := {x c(E , F ) : x(F ) 0}. F E , c0 (E ) := c0 (E , F ) , . E := N c0 := c0 (E ). c0 . , , l (E , F ). (4) S := (E , X, ) . , X RE , E X R , : X R (), . . # X+ xn 0, xn X xn (e) 0 e E . , , f F E ( S ) (, , ). Np (f ) := ( |f |p )1/p p 1, ( ). dom N1 . f F E Re f , Im f RE . , N1 (f ) = N (f ), N (g ) :=

5.5. := inf sup xn : (xn ) X, xn xn
+1

91 , ( e E ) |g (e)| = lim xn (e)
n

g F E . F = R , dom N1 X (dom N , N ). Ё N1 (f g ) Np (f )Np (g ) 1 1 = 1, p > 1 . + pp

: xy - xp yp p p (x, y R+ ),

|f |/Np (f ) |g |/Np (g ) , Np (f ) Np (g ) . Np (f )Np (g ) = 0 Ё . Lp := dom Np . |f +g |p (|f |+|g |)p 2p (|f ||g |)p = 2p (|f |p |g |p ) 2p (|f |p +|g |p ) N
p

, Np (f + g ) Np (f ) + Np (g ).

p = 1 . p > 1 Np (f ) = sup{N1 (f g )/Np (g ) : 0 < Np (g ) < +} (f Lp ), . Ё , Np (f ) > 0 g := |f |p/p g Lp , , Np (f ) = N1 (f g )/Np (g ). , N1 (f g ) = |f |p/p +1 = Np (f )p , p/p + 1 = p (1 - 1/p) + 1 = p. , Np (g )p = |g |p = |f |p = Np (f )p , Np (g ) = Np (f )p/p . N1 (f g )/Np (g ) = Np (f )p /Np (f )p/p = = Np (f )p-p/p = Np (f )p(1
-1/p )

= Np (f ),

92

. 5.

. dom N1 . f F E Re f , Im f RE . , , N (g ) := := inf sup
n

xn : (xn ) X, xn xn

+1

, ( e E ) |g (e)| lim xn (e)
n

g F E . F := R, dom N1 , , X (dom N , N ). - Lp / ker Np , - · p , , () p- , p- Lp . , Lp (S ), Lp (E , X, ) . . , S ( , A , µ), Lp ( , A , µ), Lp ( , µ) Lp (µ), . . . Lp -

. - n t := k=1 Np (fk ), fk Lp . n := k=1 fk n sn := k=1 |fk |. , (sn ) . (sp ). , sp tp < +. n n e E g (e) := lim sp (e) , n g L1 . h(e) := g 1/p (e), , h Lp sn (e) h(e) e E . |n | sn h , e E k=1 fk (e). f0 (e) |f0 (e)| h(e), , , , f0 Lp . (= ), :

5.5.
1/p

93

|n - f0 |p 0. , Np (n - f0 ) = (Lp , Np ) . 5.5.35.5.5. S E , . . , X := eE R R x := eE x(e), Lp , p- . lp (E ).
p . E := N lp x p := eE |x(e)| , p- . 1/p

(5) L . X e X+ . pe , e, [-e, e], . . pe (x) := inf {t > 0 : -te x te}. Xe , dom pe , e , e Xe . ker pe ( e). - Xe / ker pe - , e ( X ). , C (Q, R) Q , 1 := 1Q : q 1 (q Q) ( ). RE 1 l (E ). S := E , X, 1 E F, N (f ) := inf {t > 0 : |f | t1} < +, . L .

94

. 5.

- L / ker N L , · . L , , ( L ) . L . L , , C (Q, F), lp (E ), c0 (E ), c, lp , Lp (p 1), . , . . , L1 ( - ). , X , X Lp p 1. (6) S := E , X, p 1. , e E (Ye , · Ye ). f eE Ye |||f ||| : e f (e) Ye . , , Np (f ) := inf {Np (g ) : g Lp , g |||f |||}. , dom Np Np . - dom Np / ker Np |||·|||p (Ye )eE p (, Lp S ). p . k=1 Np (fk ) < +. n (sn := k=1 |||fk |||) g Np (g ) < +. , e E (sn (e)), . . k=1 fk (e) Ye . - Ye , k=1 fk (e) f0 (e) Ye e E . f0 (e) Ye g (e) e E , , n f0 dom Np . , Np ( k=1 fk - f0 ) k=n+1 Np (fk ) 0. E := N Y (Yn )nN Y := (Y1 Y2 . . . )p , p . y Y -

5.5. (yn )n
N

95

, yn Yn
1/p p n Yn

|||y |||p :=
k=1

y

< +.

, Ye := X e E , X F, Fp := dom Np Fp := Fp / ker Np . X- E ( , p- ). , Fp . , Fp (, Fp = Lp ). Fp , . , ґ Fp , , , . Fp Lp ( : Lp (X), Lp (S, X), Lp ( , A , µ), Lp ( , µ) . .) X- , p- , p- X- . , Lp (X) . p = 1. , f : f= f -1 (x) x,
ximf

f

-1

(x) x im f . , |||f ||| =
ximf

f

-1

(x)

x=

=
ximf

f

-1

(x)

x=
ximf

x

f

-1

(x)

< +.

96

. 5.

f X : f :=
ximf

f

-1

(x)

x.

, , , . , , f=
ximf

f

-1

(x)

x
ximf

f

-1

(x)

x=

=
ximf

x f

-1

(x)

=

|||f |||.

4.5.10 5.3.8 B (L1 (X), X). ( E . .) . (7) , . , . . (xn ) . ( ) ( xn ), . . (xn ). (xn ) (xn ) () . n=1 xn < + (= x1 + x2 + . . . ). 5.5.3 x = n=1 xn : x = lim s , s := ( ) , n xn N. x (xn ), (xn ) x (: x = nN xn ). : ( ). dim X < + (= ). .

5.6.

97

. X 2 (tn ) , n=1 tn < +, (xn ), xn = tn n N. (X, M) . , (xe )eE ( x) x := eE xe , x (X, M) (s ), E , . . s x (X, M). p eE p(xe ), , (xe )eE (, , , ). Y T B (X, Y). T L1 (X) L1 (Y), X- f , T f : e T f (e) e E . f L1 (X) T f L1 (Y) E T f = T E f . : . 5.6. 5.6.1. X , Y , Z , T L (X, Y ) S L (Y , Z ) . S T S T , . . . x X ST x S Tx S T x.

5.6.2. . , , () F. A F, () : (a, b) ab (a, b A). , (. . (A, +, ) () ) , , , (ab) = (a)b = a(b) a, b A

98

. 5.

F. , (A, F, +, · , ). , , A. 5.6.3. . ( ) ( ), . . 5.6.4. B (X ) := B (X, X ) X . B (X ) . X = 0 B (X ) IX IX = 1. B (X ) , X . X = 0, . X = 0, 5.5.6. 5.6.5. . 5.6.4 IX , F, . ( , 1 = I0 = 0!) X = 0 F FIX . 5.6.6. . X T B (X ). r(T ) := inf T n 1/n : n N T . ( (. 8.1.12).) 5.6.7. r(T ) T . , 5.6.1, T
n

T

n

.

5.6.8. r(T ) = lim
n

T

n

.

> 0 s N , T s (r(T ) + )s . n N n s n = k (n)s + l(n), k (n), l(n) N 0 l(n) s - 1. , T
n

=T

k (n) s

T

l (n)

T

s k (n)

T

l ( n)

5.6. 1 T ... T , r(T ) T M
1/n n 1/n s-1

99 =M T
s k ( n)

T

s k (n)

.

M

1/n

T

s k(n)/n

n-l(n))/n

(r(T ) + )k

(n)s/n

=M

1/n

(r(T ) + )(

.

M 1/n 1 (n - l(n))/n 1, r(T ) lim sup T n 1/n r(T ) + . lim inf T n 1/n r(T ) . . 5.6.9. . X T B (X ). : (1) 1 + T + T 2 + . . . B (X ); (2) T k < 1 k N; (3) r(T ) < 1. k (1)(3) = k=0 T -1 (1 - T ) . (1) (2): , (T k ) . (2) (3): . (3) (1): 5.6.8 > 0 k N r(T ) T k 1/k r(T ) + < 1. k , k=0 T . B (X ) 5.5.3, , k=0 T k B (X ). n S := k=0 T k Sn := k=0 T k . S (1 - T ) = lim Sn (1 - T ) = lim (1 + T + . . . + T n ) (1 - T ) = = lim(1 - T
n+1

) = 1;

(1 - T )S = lim(1 - T )Sn = lim(1 - T )(1 + T + . . . + T n ) = = lim 1 - T
n+1

= 1,

T n 0. , 2.2.7, S = (1 - T )-1 .

100

. 5.

5.6.10. . T < 1, (1 - T ) () (= ), . . . (1 - T )-1 (1 - T )-1 . ,

(1 - T )-1
k=0

T

k

k=0

T

k

= (1 - T )-1 .

5.6.11. . 1 - T < 1, T 1-T 5.6.9,
-1

1-T . 1- 1-T

1+

(1 - T )k =
k=1 k=0

(1 - T )k = (1 - (1 - T ))-1 = T

-1

.

:

T

-1

-1 =

(1 - T )k
k=1 k=1

(1 - T )k
k=1

1-T

k

.

5.6.12. . XY . () . T T -1 . S, T B (X, Y ) , T -1 B (Y , X ) , , T -1 S - T 1/2. T -1 S B (X ). 1-T
-1

S=T

-1

T -T

-1

ST

-1

T -S

1 < 1. 2

5.6.11, (T -1 S )-1 B (X ). R := (T -1 S )-1 T -1 . , R B (Y , X ) , , R = S -1 (T -1 )-1 T -1 = S -1 .

5.6. , S =S S S
-1 -1 -1

101

-T
-1

-1

S
-1

-1

-T

-1

= 1 S 2
-1

(T - S )T 2 T

S

T -S

T

-1

.

-1

-1

. T -S T
-1

-T

-1

S

-1

2 T

-1 2

T -S .

5.6.13. . X F T B (X ). F T , ( - T )-1 B (X ). R(T , ) := ( - T )-1 R(T , ) ( T ). res(T ). R(T , ) res(T ) B (X ) T . F \ res(T ) T Sp(T ) (T ). . 5.6.14. . X = 0, T = 0 B (X ) . , X = 0. X = 0 F := R , F := C (. 8.1.11). 5.6.15. res(T ) , 0 res(T ), 0

R(T , ) =

(-1)k ( - 0 )k R(T , 0 )k
k=0

+1

.

|| > T , res(T ) R(T , ) = 1

k=0

Tk , k

R(T , ) 0 || +.

102

. 5.

( - T ) - (0 - T ) = | - 0 |, res(T ) 5.6.12. , - T = ( - 0 ) + (0 - T ) = (0 - T )R(T , 0 )( - 0 ) + (0 - T ) = = (0 - T )(( - 0 )R(T , 0 ) + 1) = (0 - T )(1 - ((-1)( - 0 )R(T , 0 ))). , 0 5.6.9 R(T , ) = ( - T )-1 = = (1 - ((-1)( - 0 )R(T , 0 )))-1 (0 - T )-1 =

=
k=0

(-1)k ( - 0 )k R(T , 0 )k

+1

.
-1

5.6.9 || > T (1 - T /) , . . 1 R(T , ) = R(T , ) 1 1 · . || 1 - T /|| T 1-
-1

,

1 =

k=0

Tk . k

5.6.16. T . 5.6.17. . , || > r(T ) R(T , )= k=0 T k /k+1 , (. 8.1.12). 5.6.18. S T , S T . : S T R(T , )S ( - : S R(T T )S = S (0 - = T S S ( - T ) = S - S T = S - T S = ( - T )S T ) = S R(T , )S = S R(T , ) ( res(T )). , 0 ) = R(T , 0 )S S = R(T , 0 )S (0 - T ) (0 - T ) T S = ST .

5.6.

103

5.6.19. , µ res(T ), (= ) R(T , ) - R(T , µ) = (µ - )R(T , µ)R(T , ). µ - = (µ - T ) - ( - T ) R(T , ) , R(T , µ) , . 5.6.20. , µ res(T ), R(T , )R(T , µ) = R(T , µ) R(T , ). 5.6.21. res(T ) dk R(T , ) = (-1)k k ! R(T , )k dk
+1

.

5.6.22. . Sp(S T ) Sp(T S ) . , 1 Sp(S T ) 1 Sp(T S ). , Sp(S T ) = 0 1 1 Sp(S T ) 1 Sp 1 ST 1 Sp 1 TS Sp(T S ).

, 1 Sp(S T ). (1 - S T )-1 1 + S T + (S T )(S T ) + (S T )(S T )(S T ) + . . . , T (1 - S T )-1 S T S + T S T S + T S T S T S + . . . (1 - T S )-1 - 1 , (1 - T S )-1 = 1 + T (1 - S T )-1 S

104

. 5.

( 1 Sp(T S )). : (1 + T (1 - S T )-1 S )(1 - T S ) = = 1 + T (1 - S T )-1 S - T S + T (1 - S T )-1 (-S T )S = = 1 + T (1 - S T )-1 S - T S + T (1 - S T )-1 (1 - S T - 1)S = = 1 + T (1 - S T )-1 S - T S + T S - T (1 - S T )-1 S = 1; (1 - T S )(1 + T (1 - S T )-1 S ) = = 1 - T S + T (1 - S T )-1 S + T (-S T )(1 - S T )-1 S = = 1 - T S + T (1 - S T )-1 S + T (1 - S T - 1)(1 - S T )-1 S = = 1 - T S + T (1 - S T )-1 S + T S - T (1 - S T )-1 S = 1 , .
5.1. , , . 5.2. , . 5.3. , , . 5.4. , . 5.5. RN N ? 5.6. , . 5.7. . ? 5.8. . 5.9. . 5.10. .

105

5.11. lp lq , Lp Lq . ? 5.12. , , . 5.13. , C.

5.14. , ( ) . 5.15. ? 5.16. F C (Q, F), Q ? 5.17. , Lp (X ) = Lp (X ), X . 5.18. (Xn )

X0 :=

x
nN

Xn : x

n Xn

0

c0 ( x := sup{ xn : n N}, l ). , X0 , Xn . 5.19. , C (p) [0, 1] , C [0, 1].

6

6.1. 6.1.1. . H F. f : H 2 F , (1) f (· , y ) : x f (x, y ) H # y Y ; (2) f (x, y ) = f (y , x) x, y H , F, . . . 6.1.2. . , f # x H f (x, · ) : y (x, y ) H , H H (. 2.1.4 (2)). , F := R , . . , F := C , . . - . 6.1.3. f : f (x + y , x + y ) - f (x - y , x - y ) = 4 Re f (x, y ) (x, y H ).

-

f (x + y , x + y ) = f (x, x) + f (x, y ) + f (y , x) + f (y , y ) f (x - y , x - y ) = f (x, x) - f (x, y ) - f (y , x) + f (y , y ) 2(f (x, y ) + f (y , x))

6.1.

107

6.1.4. . f , , f (x, x) 0 x H . : (x, y ) := x | y := f (x, y ) (x, y H ). , (x, x) = 0 x = 0 (x H ). 6.1.5.

|(x, y )|2 (x, x)(y , y ) (x, y H ). (x, x) = (y , y ) = 0, 0 (x + ty , x + ty ) = t(x, y ) + t (x, y ). t := -(x, y ), -2|(x, y )|2 0, . . . , , (y , y ) = 0,

0 (x + ty , x + ty ) = (x, x) + 2t Re(x, y ) + t2 (y , y ) (t R) : Re(x, y )2 (x, x)(y , y ). (x, y ) = 0, . (x, y ) = 0, := |(x, y )| (x, y )-1 x := x. || = 1 , , (x, x) = (x, x) = (x, x) = ||2 (x, x) = (x, x); |(x, y )| = (x, y ) = (x, y ) = (x, y ) = Re(x, y ). , |(x, y )|2 = Re(x, y )2 (x, x)(y , y ). 6.1.6. ( · , · ) H , · : x (x, x)1/2 H . . , x+y
2

= (x, x) + (y , y ) + 2 Re(x, y ) y = ( x + y )2 .

(x, x) + (y , y ) + 2 x

6.1.7. . H (· , ·) · . H , (H, · ) .

108

. 6.

6.1.8. H x+y
2

+ x-y

2

= 2( x

2

+ y 2 ) (x, y H )

. x+y x-y
2 2

= (x + y , x + y ) = x = (x - y , x - y ) = x

2 2

+ 2 Re(x, y ) + y 2 ; - 2 Re(x, y ) + y
2

6.1.9. . (H, · ) , H , . . , , (· , ·) H , x = (x, x)1/2 x H . HR H x, y HR (x, y )R := 1 4 x+y
2

- x-y

2

.

, (· , y )R (x1 , y )R + (x2 , y )R = 1 4 1 = 4 = = x1 + y x1 + y 1 4 1 2
2

-

x1 - y
2

2

+ x2 + y x1 - y
2

2

- x2 - y + x2 - y
2

2

= =

2

+ x2 + y

-

2

2

1 ( (x1 + y ) + (x2 + y ) 2 (x1 - y ) + (x2 - y ) 1 x1 + x2 + 2y 2
2 2

+ x1 - x2

-

- =

+ x1 - x2 2 ) =
2

1 4

-

1 x1 + x2 - 2y 2

=

6.1. = 1 2 (x1 + x2 )/2 + y
2

109
2

- (x1 - x2 )/2 - y

=

= 2 ((x1 + x2 )/2, y )R . , x2 := 0 (x2 , y )R = 0, . . 1/2(x1 , y )R = (1/2 x1 , y )R . x1 := 2x1 x2 := 2x2 (x1 + x2 , y )R = (x1 , y )R + (x2 , y )R . (· , y )R , (· , y )R (HR )# . (x, y ) := Re
-1

((· , y )R )(x),

Re-1 (. 3.7.5). F := R , (x, y ) = (x, y )R = (y , x) (x, x) = x 2 , . . . F := C, (x, y ) = (x, y )R - i(ix, y )R . , (y , x) = (y , x)R - i(iy , x)R = (x, y )R - i(x, iy )R = = (x, y )R + i(ix, y )R = (x, y ) , (x, iy )R = 1 |i| y - ix 4 , = 1 4
2

x + iy

2

- x - iy
2

2

=

- |-i| ix + y

= -(ix, y )R .

(x, x) = (x, x)R - i(ix, x)R = = x2 - =x
2

i ix + x 2 - ix - x 2 = 4 i 1- |1 + i|2 - |1 - i|2 = x 2. 4

6.1.3.

110

. 6.

6.1.10. . (1) L2 ( - ). : (f , g ) := f g f , g L2 . , l2 (E ) (x, y ) := eE xe ye x, y l2 (E ). (2) H (· , ·) : H2 F H . , HR (· , ·)R : (x, y ) Re(x, y ) , H , H HR . (HR , (· , ·)R ) (H, (· , ·)). , , . (3) H H H . x, y H (x, y ) := (x, y ) . , (· , ·) H . H H . (4) H H0 := ker · · H . , 2.3.8 6.1.10 (3), , H/H0 : x1 := (x1 ) x2 := (x2 ), x1 , x2 H : H H/H0 , (x1 , x2 ) := (x1 , x2 ). H/H0 - (H, · ) · . , H/H0 , , H . H/H0 , (, ). . (5) (He )eE H 2, . . h H

6.2. , h := (he )e
1/2 E

111 , he He e E ,

h :=
eE

h

e

2

< +.

5.5.9 (6), H . f , g H , , 1 2 = 1 2 =
eE

f +g

2

+ f -g
2

2

=
2

fe + ge
eE

+
eE

fe - ge
2

=

1 2 fe

fe + ge
2

2

+ fe - ge =f
2

=

=
eE

+ ge

2

+ g 2,

, , H . H (He )eE eE He . E := N H := H1 H2 . . . . (6) H S . L2 (S, H ), H - , , . 6.2. 6.2.1. U (r + )BH \ rBH , 0 < r, H . : diam U 12r. x, y U , , 1/2(x + y ) U , , x-y
2

=2

x

2

+y

2

- 4 (x + y )/2

2

112

. 6. 4(r + )2 - 4r2 = 8r + 42 12r.

6.2.2. . U H x H \ U . , , u0 U , x - u0 = inf { x - u : u U }. U := {u U : x - u inf U - x + }. 6.2.1 (U )>0 U . 6.2.3. . u0 , 6.2.2, x U x U . 6.2.4. H0 H x H \ H0 . x0 H0 x H0 , (x - x0 , h0 ) = 0 h0 H0 . (H0 )R H0 . (H0 )R f (h0 ) := (h0 - x, h0 - x). x0 H0 x H0 , 0 x0 (f ). 3.5.2 (4) , (x - x0 , h0 ) = 0 h0 H0 , f (x0 ) = 2(x0 - x, ·). 6.2.5. . x, y H x y , (x, y ) = 0. U , U , . . U := {y H : x U x y }. U U . 6.2.6. H0 H . H0 , H = H0 H0 .
H0 H . , H0 H0 = H0 H0 = 0. , H0 H0 = H0 + H0 = H . h H \ H0 . 6.2.2 h0 H0 , , 6.2.4, h - h0 H0 . , h = h0 + (h - h0 ) H 0 + H 0 .

6.2.

113

6.2.7. . H0 H0 H0 PH0 . 6.2.8. . x y x + y
2

=x

2

+ y 2.

6.2.9. . : H = 0, H0 = 0 PH0 = 1. 6.2.10. . P L (H ) , P 2 = P , : (1) P H0 := im P ; (2) h 1 P h 1; (3) (P x, P d y ) = 0, P d := IH - P x, y H ; (4) (P x, y ) = (x, P y ) x, y H . (1) (2): 6.2.9. (2) (3): H1 := ker P = im P d . x H1 . d d 2 2 d x = P x + P x x P x, x P x = (x - P x, x - P d x) = (x, x) - 2 Re(x, P d x) + (P d x, P d x) = x 2 + P d x 2 . P d x = 0, . . x im P . H1 = ker P H1 im P d 6.2.6 : H1 = im P = H0 . , (P x, P y ) = 0 x, y H , P x H0 , P d y H1 . (3) (4): (P x, y ) = (P x, P y + P d y ) = (P x, P y ) = (P x, P y ) + d (P x, P y ) = (x, P y ). (4) (1): , H0 . h0 := lim hn hn H0 , . . P hn = hn . x H (· , x) (· , P x) (h0 , x) = lim (hn , x) = lim (P hn , x) = lim (hn , P x) = (P h0 , x). (h0 P x, h0 ) = (P x - P x, P x = PH0 x. - P h0 , h0 - P h0 ) = 0, . . h0 im P . x H h0 H0 (x - (x - P x, P h0 ) = (P (x - P x), h0 ) = (P x - P 2 x, h0 ) = h0 ) = 0. , 6.2.4,

6.2.11. P1 , P2 , P1 P2 = 0. P2 P1 = 0. P1 P2 = 0 im P2 ker P1 im P1 = (ker P1 ) (im P2 ) = ker P2 P2 P1 = 0

114

. 6.

6.2.12. . P1 P2 ( P1 P2 P2 P1 ), P1 P2 = 0. 6.2.13. . P1 , . . . , Pn . P := P1 + . . . + Pn , Pl Pm l = m. : , 2 P0 6.2.10 P0 x 2 = (P0 x, P0 x) = (P0 x, x) = (P0 x, x). , x H l = m Pl x
n n 2

+ Pm x

2

2

k=1

Pk x

2

=
k=1

(Pk x, x) = (P x, x) = P x

x 2.

, x := Pl x, Pl x
2

+ Pm Pl x

2

Pl x

2

Pm P

l

= 0.

: , P . ,
n 2 n n 2

-

n

P=
k=1

Pk

=
l=1 m=1

Pl Pm =
k=1

2 Pk = P.

, 6.2.10 (4), (Pk x, y ) = (x, Pk y ) , , (P x, y ) = (x, P y ). 6.2.10 (4). 6.2.14. . 6.2.13 . 6.3. 6.3.1. . (xe )eE H , e1 = e2 xe1 xe2 . E H , (e)eE .

6.3.

115

6.3.2. . (xe )eE () , ( xe 2 )eE .
2

xe
eE

=
eE

xe 2 .

s := eE xe , E . 6.2.8, s 2 = e xe 2 . , , , s - s
2

= s

\

2

=
e \

xe 2 .

, (s ) ( xe 2 )eE . 5.5.3, . 6.3.3. . (Pe )eE H . x H () (Pe x)eE . P x := eE Pe x H :=
eE

xe : xe He := im Pe ,
eE

xe

2

< + .

E s := e Pe . 6.2.13, s . , 6.2.8, s x 2 = Pe x 2 x 2 x H . e , ( Pe x 2 )eE ( ). P x := eE Pe x, . . P x = lim s x. P 2 x = lim s P x = lim s lim s x = lim lim s s x = lim lim s x= lim s x = P x. P x = lim s x = lim s x x , , P 2 = P . 6.2.10, , P im P . x im P , . . P x = x, x = eE Pe x eE Pe x 2 = x 2 = P x 2 < +. Pe x

116

. 6.

2 He (e E ), x H . xe He < + , eE xe x := eE xe ( ) x = eE xe = eE Pe xe = P x, . . x im P . , im P = H .

6.3.4. . H (He )eE . , , , . 6.3.5. . h H : h = 1. , , H0 := Fh H , h0 . x H F (x - (x, h)h, h) = ((x, h) - (x, h))(h, h) = 0. , 6.2.4, PH0 = (· , h) h. h . , h : x (x, h)h (x H ). 6.3.6. . ( ), , -, , -, . . 6.3.7. E H x H ( e x)eE () . : x
2

eE

|(x, e)|2 .

,
2 2 2

x

eE

ex

=
eE

(x, e)e

=
eE

(x, e)e

2

.

6.3.

117

6.3.8. . E H ( H ), x H x = eE e x. , . 6.3.9. E H , L (E ) H . 6.3.10. . , E , E = 0. 6.3.11. . , . : h E . h = eE e h = eE (h, e)e = 0 = 0. eE : x H , 6.3.3 6.2.4, x - eE e x E . 6.3.12. . . H E . h H \ H0 , H0 := cl L (E ), h1 := h - PH0 h E , , H = 0 E { h1 -1 h1 } = E . . H = 0 . 6.3.13. . , H . H . 6.3.14. . (xn )nN H . x0 := 0, e0 := 0,
n-1

yn := xn -
k=0

ek xn ,

en :=

y y

n n

(n N).

118

. 6.

, (yn , ek ) = 0 0 k n - 1 (, 6.2.13). , yn = 0, H . (en )nN , , , (xn )nN . , , , , . . . 6.3.15. . E H x H . x := (xe )eE F E , xe := (x, e), x ( E ). 6.3.16. . E H . F : x x ( E ) H l2 (E ). F -1 : l2 (E ) H -1 F (x) := eE xe e x := (xe )eE l2 (E ). x, y H (x, y ) =
eE xe ye .

l2 (E ). 6.3.3, . , . , F -1 x = x x H F -1 (x) = x x l2 (E ), . x
2

=
eE

xe

2

=x

2 2

(x H )

. (x, y ) =
eE

xe e,
eE

ye e

=
e,e E

xe ye (e, e ) = eE

xe ye .

6.3.17. . , . ,

6.4.

119

, . . , . ( ) . 6.4. 6.4.1. . H . x H x := (· , x). x x H H . , x = 0 x = 0. x = 0, x x
H

= sup |(y , x)| sup
y 1 y 1

y

x x;

H

= sup |(y , x)| |(x/ x , x)| = x .
y 1

, x x H H . , . l H H0 := ker l = H ( l , ). e = 1 , e H0 , grad l := l(e) e. x H0 , (grad l) (x) = (x, grad l) = (x, l(e) e) = l(e) (x, e) = 0. , F x H 2.3.12 (grad l) (x) = l(x). , x := e (grad l) (e) = (e, grad l) = l(e)(e, e) = l(e), . . = 1. 6.4.2. . , H x x H H . l grad l. 6.4.1 .

120

. 6.

6.4.3. . : H H , . . H H , x (l) = (x)(l) = l(x), x H l H (. 5.1.10 (8)). , . f H . y f (y ) y H . , H , , x H = H , (y , x) = (x, y ) = f (y ) y H . (x)(y ) = y (x) = (x, y ) = f (y ) y H . y y H , (x) = f . 6.4.4. H1 , H2 T B (H1 , H2 ). , , T : H2 H1 , x H1 , y H2 (T x, y ) = (x, T y ). T B (H2 , H1 ) T = T . y H2 . x (T x, y ) y T , . . H1 . x H1 , x = y T . T y := x. , T L (H2 , H1 ). , , |(T y , T y )| = |(T T y , y )| T T y yT

T y

y.

, T y T y y H2 , . . T T = T := (T ) , . . T = T T .

T .

6.4.5. . T B (H2 , H1 ), 6.4.4, T B (H1 , H2 ). 6.4.6. H1 , H2 , , S, T B (H1 , H2 ) F. (1) T = T ; (2) (S + T ) = S + T ; (3) (T ) = T ; (4) T T = T 2 .

6.4. (1)(3) Tx
2

121

. x 1, = (T x, T x) = |(T x, T x)| = |(T T x, x)| T T x x T T .

, 6.4.4, T T T T = T 2 , (4). 6.4.7. H1 , H2 , H3 , T B (H1 , H2 ) S B (H2 , H3 ). (S T ) = T S . (S T x, z ) = (T x, S z ) = (x, T S z ) (x H1 , z H3 ) 6.4.8. . T T H1 H2 . H1 - H2 . , , , . 6.4.9. . , . 6.4.7 6.4.6 (1). 6.4.10. . T B (H1 , H2 ) T B (H2 , H1 ). T , T . T -1 = T -1 . 6.4.11. . T B (H ) Sp(T ) Sp(T ). 6.4.12. (. 7.6.13). . . . - H
k-1
k - Hk - H

T

Tk

+1

k+1

- . . .

, . . . - H
k-1
k - Hk - H

T

Tk+1

k+1

- . . . .

122

. 6.

6.4.13. . - ( F) A , . . a a A , (1) a = a (a A); (2) (a + b) = a + b (a, b A); (3) (a) = a ( F, a A); (4) (ab) = b a (a, b A). A , a a = a 2 a A, C -. 6.4.14. B (H ) H C - ( ). 6.5. 6.5.1. . H F T B (H ). T ( ), T = T . 6.5.2. . T T = sup |(T x, x)|.
x 1

Tx x . . (x, y )

t := sup{|(T x, x)| : x 1}. , T , x 1. , t T = T , (T x, y ) = (x, T y ) = (T y , (T x, y ) . ,
2

|(T x, x)| T. x) = (y , T x) , 6.1.3 6.1.8

4 Re(T x, y ) = (T (x + y ), x + y ) - (T (x - y ), x - y ) t( x + y + x - y 2 ) = 2t( x
2

+ y 2 ).

T x = 0, T x t. T x = 0. x 1 y := T x -1 T x Tx = Tx = (T x, y ) = Re(T x, y ) . . T = sup{ T x : Tx , Tx
2

Tx Tx

=
2

1 tx 2 x 1} t.

+ T x/ T x

t,

6.5.

123

6.5.3. . 6.5.2, T H fT (x, y ) := (T x, y ). , , f , y H f (· , y ) . T y H , f (·, y ) = (T y ) . , T L (H ) (x, T y ) = f (x, y ) = f (y , x) = (y , T x) = (T x, y ). , T B (H ) T = T . , f = fT . , 6.5.1 T B (H ) T L (H ) ( Ё , . 7.4.7). 6.5.4. . T , inf
x =1

x - T x = 0.

: t := inf { x - T x : x H, x = 1} > 0. , Sp(T ). x H x - T x t x . / , -, ( - T ) , -, H0 := im(-T ) ( (-T )xm -(-T )xk t xm - xk , . . ) , , -, ( - T )-1 B (H ), H = H0 ( R(T , ) t-1 ). , , H = H0 . y H0 , y = 1. x H 0 = (x - T x, y ) = (x, y - T y ), . . y = T y . , = (T y , y )/(y , y ) T R. = y ker( - T ). : 1 = y = 0 = 0. : Sp(T ), R(T , ) B (H ). / inf { x - T x : x = 1} R(T , ) -1 . 6.5.5. . T . mT := inf (T x, x),
x =1

-

MT := sup (T x, x).
x =1

Sp(T ) [mT , MT ] mT , MT Sp(T ).

124

. 6.

T - Re H , x - T x
2

= | Im |2 x

2

+ T x - Re x

2

6.5.4 Sp(T ) R. < mT , x H x = 1 6.1.5 x - T x = x - T x x |(x - T x, x)| =

= | - (T x, x)| = (T x, x) - mT - > 0. 6.5.4 : res(T ). > MT , x - T x |(x - T x, x)| = | - (T x, x)| = - (T x, x) - MT > 0. res(T ). Sp(T ) [mT , MT ]. (T x, x) R x H , 6.5.2 T = sup{|(T x, x)| : = sup{(T x, x) (-(T x, x)) : x 1} =

x 1} = MT (-mT ).
2 2

, := T = MT . x = 1, x - T x
2

= 2 - 2(T x, x) + T x

2 T

- 2 T (T x, x).

, inf
x =1

x - T x

2

2 T

inf ( T - (T x, x)) = 0.
x =1

6.5.4, : Sp(T ). S := T - mT . , MS = MT - mT 0 mS = mT - mT = 0. , S = MS MS Sp(S ). , MT Sp(T ), T = S + mT , MT = MS + mT . , mT = -M-T Sp(T ) = - Sp(-T ). 6.5.6. . ( ). 6.5.7. . , .

6.6. 6.6.

125

6.6.1. . X Y . T L (X, Y ) ( T K (X, Y )), T (BX ) BX X Y . 6.6.2. . . . 8. 6.6.3. T . 0 = Sp(T ), T , . . ker( - T ) = 0. (xn ) , xn = 1, xn - T xn 0. , , (T xn ) y := lim T xn . xn = (xn - T xn ) + T xn , (xn ) y = lim xn . , T y = T (lim xn ) = lim T xn = y . y = ||, , y T . 6.6.4. 1 , 2 T , x1 , x2 1 2 (. . xs ker(s - T ), s := 1, 2). x1 x2 . (x1 , x2 ) = 1 1 2 (T x1 , x2 ) = (x1 , T x2 ) = (x1 , x2 ) 1 1 1

6.6.5. > 0 [-, ] . (n )nN T , |n | > . , , xn , n , xn = 1. 6.6.4 (xk , xm ) = 0 m = k . , T xm - T xk
2

= T xm

2

+ T xk

2

= 2 + 2 22 , m k

. . (T xn )nN . T .

126

. 6.

6.6.6. . T H 0 = Sp(T ). H := ker( - T ). H H = H H T . 0 T , 0 T T T H Sp(T ) = Sp(T ) \ {}.

,

H T . , H T . , H H T (= T ), ( x H )(x, h) = 0 ( x H )(T h, x) = (h, T x) = 0. T H . T T H . , µ = µ-T µ- 0 0 µ-T

, µ - T . , T . 6.6.7. . H T H . , , P ker( - T ) Sp(T ). T= P .
Sp(T )

6.5.6 6.6.6, Sp(T ) T-

P

= sup{|| : (Sp(T ) 0) \ }.

6.6.5.

6.6.

127

6.6.8. . , T , . . , , H0 (H0 := ker T ). , T , . . , H0 , ,
n n

T=
k=1

k ek =
k=1

k ek ek ,

1 , . . . , n T , , {e1 , . . . , en } H0 , . , , . , = µ, , µ T , H Hµ . Sp(T )\0 H H0 = cl im T , H0 = (im T ) . H ( , . . 1 := 2 := . . . := dim H1 := 1 ; dim H1 +1 := . . . := dim H1 +dim H2 := 2 . .), H = H0 H1 H2 . . .

T=
k=1

k ek =
k=1

k ek ek ,

. 6.6.9. . T K (H1 , H2 ) , H1 H2 . (ek )kN H1 , (fk )kN H2 (µk )kN R+ \ 0, µk 0,

T=
k=1

µk ek fk .

128

. 6.

S := T T . , S B (H1 ) S . , (S x, x) = (T T x, x) = (T x, T x) = T x 2 . , 6.4.6, S H0 := ker S = ker T . , Sp(S ) R+ 6.5.5. (ek )kN H0 S (k )kN k > 0, k N (. 6.6.8). x H1

x - PH0 x =
k=1

(x, ek )ek . = 0, µk :=

, , T P fk := µ-1 T ek , k

H0

k

Tx =
k=1

(x, ek )T ek =
k=1 N

(x, ek )

µk T ek = µk

µk (x, ek )fk .
k=1

(fk )k

, T en T em , µn µm = 1 (T en , T em ) = µn µm

(fn , fm ) =

=

1 1 (T T en , em ) = (S en , em ) = µn µm µn µm = 1 µn (n en , em ) = (en , em ). µn , µm µm

, :
n 2 2

T-
k=1

µk ek fk

x

=
k=n+1

µk (x, ek )fk

=

=
k=n+1

µ2 |(x, ek )|2 k

n+1 k=n+1

|(x, ek )|2

n+1

x 2.

129

, k 0,
n

T-
k=1

µk ek fk µn

+1

0.

6.6.10. . 6.6.9 , , ( ) . : .
6.1. . 6.2. , , . 6.3. - ? 6.4. ? 6.5. ? 6.6. . 6.7. , . 6.8. L2 n + 1 n. x
2

6.9. , x y , x + y + y 2 x + iy 2 = x 2 + y 2 . 6.10. T (ker T )

2

=

= cl im T ,

(im T )

= ker T .

6.11. . 6.12. , , . 6.13. , , . ?

130

. 6.
.

6.14. , T T ?

6.15. , . ? 6.16. ? 6.17. , . 6.18. , P , P P . 6.19. (akl )k,lN , a k, l , , pk , > 0 ,
kl

0

akl pk pl ;
k=1 l=1

akl pl p

k

(k , l N).

T B (l2 ) , (ek , el ) = akl T = ( ek l2 , N).

7

7.1. 7.1.1. . U () : int U = . (1) 0 < 1 cl U + (1 - ) int U int U ; (2) core U = int U ; (3) cl U = cl int U ; (4) int cl U = int U . (1) u0 int U 5.2.10 int U - u . 0 < 1 cl U cl U U + (1 - )(int U - u0 ) = = U + (1 - ) int U - (1 - )u0 U + (1 - )U - (1 - )u0 U - (1 - )u0 . , (1 - )u0 + cl U U , , U (1 - ) int U + cl U . , cl U (1 - ) int U . (2) , int U core U . u0 int U u core U , u1 U 0 < < 1 u = u0 + (1 - )u1 . u1 cl U , (1) : u int U .
0

-

132

. 7.

(3) , cl int U cl U , int U U . , , u cl U , , u0 int U u := u0 +(1-)u, : u u 0 u int U , 0 < < 1. , u cl int U . (4) int U U cl U , int U int cl U . u int cl U , , (2), u core cl U . , u0 int U , u1 cl U 0 < < 1, u = u0 + (1 - )u1 . (1), : u int U . 7.1.2. . c int U = 7.1.1 (2) 7.1.1 (4) . , , . , U := Bc0 X , c0 , X c0 , . . . , , core U = cl U = Bc0 . 7.1.3. . U () X , U . , U , , (n )nN (un )nN , n R+ , n=1 n = 1 un U , n=1 n un X u, u U . 7.1.4. . (1) ( u0 ) x x + u0 . (2) . (3) . , U . U = , . U = , 7.1.4 (1) , 0 U , , U = {pU < 1}, pU U . (un )nN (n )nN U R+ , n=1 n = 1 u := n=1 n un U . 7.1.4 (2), u cl U = {pU 1} , , pU (u) = 1. , , pU (u) n=1 n pU (un ) 1 = n=1 n (. 7.2.1). , 0 =

7.1.
n=1

133

(n - n pU (un )) = n=1 n (1 - pU (un )). n = 0 n N. . (4) . (5) . 7.1.5. . . U . X X = nN n cl U . X , , n N, int n cl U = . , int cl U = 1/n int n cl U = . , 0 core cl U . , 7.1.1 : 0 int cl U . , > 0 , cl U BX . , : 1 > 0 cl U BX . , U /2 BX . x0 /2 BX . := 2, y1 1/ U y1 - x0 1/2 . u1 U , 1/2 u1 - x0 1/2 = 1/4 . x0 := -1/2 u1 + x0 := 4 , u2 U , 1/4 u2 + 1/2 u1 - x0 1/2 = 1/8 . , (un )nN U , , n=1 1/2n un n x0 . = 1 U n=1 1/2 , : x0 U. 7.1.6. , , . , int U core U core cl U . u core cl U , cl(U - u) = cl U - u . (. 7.1.4 (1)). , U - u 7.1.5. 5.2.10, u

134

. 7.

int U . , int U = core U = core cl U . 7.1.1, int cl U = int U. 7.1.7. . . 7.1.8. . 7.1.5 , 7.1.7 . , . , , . , , . . , . 7.1.9. . , . , , , f . {|f | 1} . 7.2. 7.2.1. p : X R (X, · ). : (1) p ; (2) p ; (3) p ; (4) {p 1} ; (5) p := sup{|p(x)| : x 1} < +, . . p . (1) (2) (3) (4) . (4) (5): t > 0, t-1 BX {p 1}. x 1 p(x) t. , -p(-x) p(x) , -p(x) t x BX . p t < +. (5) (1): p x, y X p(x) - p(y ) p(x - y ); p(y ) - p(x) p(y - x).

7.2. |p(x) - p(y )| p(x - y ) p(y - x) p x-y .

135

7.2.2. . , , . p . {p 1} (. 4.3.8). dom p , , 3.8.8, {p 1} . {p 1} . 7.2.1. 7.2.3. . : X , 7.2.1 (1)7.2.1 (5) : p . , dom p = X , dom p , X . 7.2.4. . X Y () . E X Y : (1) E , . . x X Y {T x : T E }; (2) E . (1) (2): q (x) := sup{p(T x) : T E }, p Y . , q , , q < +, . . p(T (x - y )) q x - y T E . , T в-1 ({dp }) {d · / q } T E , > 0 . E . (2) (1): . 7.2.5. . X Y . (T ) : (1) x X sup T x < +; (2) sup T < +.

136

. 7. , 7.2.5 (2) 7.2.4

(2). 7.2.6. X U X . : (1) U X ; (2) x X { x | x : x U } F. 7.2.5. 7.2.7. X U X . : (1) U X ; (2) x X { x | x : x U } F. (2) (1). X (. 5.5.7), X X (. 5.1.10 (8)), 7.2.6. 7.2.8. . 7.2.7 (2) : U (X, (X, X )) , 5.1.10 (4), : U . 7.2.6 7.2.7 10.4.6. 7.2.9. . X, Y (Tn )nN , Tn B (X, Y ), . E := {x X : lim Tn x}. : (1) E = X ; (2) supnN Tn < + E X . (1), (2) T0 : X Y , T0 x := lim Tn x, T0 lim inf Tn . E = X , , , cl E = X . , x X (Tn x)nN Y ( ). , supnN Tn < + (1) (2) .

7.2.

137

(2) x X , x E m, k N Tm x - Tk x = Tm x - Tm x + Tm x - Tk x + Tk x - Tk x Tm x - Tm x + Tm x - Tk x + Tk x - Tk x T
m

x - x + Tm x - Tk x + T
n nN

k

x-x

2 sup T

x - x + T m x - Tk x .

> 0 , -, x E , 2 supn Tn x - x /2, -, n N , Tm x - Tk x /2 m, k n. Tm x - Tk x , . . (Tn x)nN Y . Y , : x E . , (2) (1) . , x X T0 x = lim Tn x lim inf . T
n

x,

7.2.10. . 7.2.9 (1) 7.2.9 (2) , (Tn ) T0 X . , () Q X sup
xQ

Tn x - T0 x 0.

, pn (x) := sup{ Tm x - T0 x : m n} . pn (x) pn+1 (x) pn (x) 0 x X . , : , , .

138

. 7. X (Tn )nN Tn = +, Tn x = +. .

7.2.11. . Y . B (X, Y ) supn x X , supn

. 7.2.3 4.7.4. 7.2.12. . X Y . (Tn,m )n,mN B (X, Y ) , supn Tn,m = + m N, x X , supn Tn,m x = + m N. 7.3. 7.3.1. X Y . F X в Y , x1 , x2 X 1 , 2 R+ , 1 + 2 = 1, F (1 x1 + 2 x2 ) 1 F (x1 ) + 2 F (x2 ). : (x1 , y1 ), (x2 , y2 ) F 1 , 2 0, 1 + 2 = 1, 1 y + 2 y2 F (1 x1 + 2 x2 ), y1 F (x1 ) y2 F (x2 ). : x1 x2 dom F , . x1 , x2 dom F y1 F (x1 ), y2 F (x2 ), 1 (x1 , y1 ) + 2 (x2 , y2 ) F 1 , 2 0, 1 + 2 = 1 (. 3.1.2 (8)). 7.3.2. . X, Y . , X в Y , X Y . , (x, y ) := x X + y Y , . . X в Y X Y 1. , , . . ( , ()). .

7.3.

139

7.3.3. . F X в Y , X Y , , , , , F . 7.3.4. . . F X в Y U X . U dom F = , F (U ) = . , (yn )nN F (U ), . . yn F (xn ), xn U n N. , , (n ) , n = 1 , , Y y := n=1 n=1 n yn . ,

n xn =
n=1 n=1

n xn
n=1

n sup U = sup U < +

U . X , 5.5.3 X x := n=1 n xn . , X в Y

(x, y ) =
n=1

n (xn , yn ).

F U , : (x, y ) F x U . , y F (U ). 7.3.5. . X Y , F X в Y (x, y ) F . F x y , y core F (X ). : . : 7.1.4 : x = 0 y = 0. U BX > 0, U := BX . U , 7.3.4, F (U ) . , F (U ) 7.1.6.

140

. 7.

y Y . , 0 core F (X ), R+ , y F (X ). , x X y F (X ). x 1, . x > 1, := x -1 < 1. , 7.3.1, : y = (1 - )0 + y (1 - )F (0) + F (x) F ((1 - )0 + x) = F (x) F (BX ) = F (U ). , x = 1, . . x BX . 7.3.6. . F , 7.3.5, F (x, y ). 7.3.7. . , 7.1.5. . 7.3.5 , , y core cl F (X ), F (X ). 7.3.4 , dom F . 7.3.5 7.1.5 . 7.3.5 . 7.3.8. . X Y F X вY . F , F . 7.3.9. . . 7.3.10. F , (xn ) X (yn ) Y , xn dom F, yn F (xn ) xn x, yn y , x dom F y F (x). 7.3.11. X Y F X в Y . , , (x, y ) F y core im F . F x y . , Ё 7.3.5.

7.4.

141

7.3.12. . F X в Y , X Y . 7.3.13. . X, Y F X в Y , im F . F . U X . y F (U ), x U , (x, y ) F . , y core im F . 7.3.5, F (U ) y , U x. , F (U ) . 7.4. 7.4.1. . T L (X, Y ) , T B (X, Y ) T . 7.4.2. X , Y T X Y . im T = Y Y . , im T = Y , . , T , T -1 L (Y , X ). - T T -1 B (Y , X ), Y ( ). coim T := X/ ker T , -. 5.5.4, coim T . , 2.3.11 T T coim T . - 5.1.3, , T . . , im T = im T = Y . 7.4.3. . T : coim T Y T , T = T . 7.4.4. . .

142

. 7.

T B (X, Y ) im T = Y . T , . 7.4.5. . X, Y T B (X, Y ). T X Y , . . ker T = 0 im T = Y , T -1 B (Y , X ). 7.4.4. 7.4.6. . 7.4.5 : . , : T x = y , T B (X, Y ), X, Y , , x y . 7.4.7. . X, Y T L (X, Y ) . T . T
-1

, T

-1

(Y ) = X .

7.4.8. . X, Y T L (X, Y ). : (1) T B (X, Y ); (2) (xn )nN X x X , xn x T xn y , y Y , y = T x. (2) T . 7.4.9. . X1 X ( ), X1 , , X2 , X = X1 X2 (. . X1 X2 = 0, X1 X2 = X ). 7.4.10. . X1 X : (1) X1 ; (2) X1 , . . P B (X ) , P 2 = P im P = X1 .

7.4.

143

(1) (2): P X X1 X2 (. 2.2.9 (4)). (xn )nN X xn x, P xn y . , P xn X1 n N. X1 , 4.1.19, y X1 . (xn - P xn X2 n N) , x - y X2 . , P (x - y ) = 0. , y = P y , . . y = P x. 7.4.8. (2) (1): , X1 = im P . (xn )nN X1 , xn x X . P xn P x P . P xn = xn , xn im P , P . x = P x, . . x X1 , . 7.4.11. . (1) . (2) c0 l . X := l (Q) Y := c0 (Q), Q . t R t (tn ) , tn t. Qt := {tn : n N}. , Qt Qt t = t . t , Qt - X/ Y V := {t : t R}. t = t t = t , V . f (X/ Y ) Vf := {v V : f (v ) = 0}. , Vf = nN Vf (n), Vf (n) := {v V : |f (v )| 1/n}. m N, v1 , . . . , vm Vf (n) , v1 , . . . , vm Vf (n) n k := |f (vk )|/f (vk ), x = k=1 k vk x 1 f m m |f (x)| = | k=1 k f (vk )| = | k=1 |f (vk )|| m/n. , Vf (n) . , Vf . , F (X/ Y ) v V , ( f F ) f (v ) = 0. q : x x(q ) (q Q) l (Q), . . ( q Q) q (x) = 0 x = 0 x l (Q). . (3) ( 6.2.6). ,

144

. 7.

X , dim X 3, P P 1, X (= ). : . , , ( ) . 7.4.12. XA = B . X, Y , Z ; A B (X, Y ), B B (Y , Z ). , , im A Y . A X @ @ B@ -Y X @ R @? Z

X B (Y , Z ) , ker A ker B . . im A = Y X0 L (Y , Z ) , X0 A = B , . , U Z X0-1 (U ) = A(B -1 (U )). B -1 (U ) B , A(B -1 (U )) . X0 B (im A, Z ) X X0 P , P - Y im A. . 7.4.13. . Z . 7.4.14. AX = B . X, Y , Z , A B (Y , X ), B B (Z, X ). , , ker A Y .

7.4. A X I @ @ B@ @ Z Y 6 X

145

X B (Z, Y ) , im A im B . . Y ker A Y0 , Y0 . 5.5.9 (1), Y0 . A0 A Y0 . , im A0 = im A im B . , 2.3.13 2.3.14 A0 X0 = B , , X0 := A-1 B . , X0 , 0 L (Z, Y0 ), . X0 . (. 7.4.8), zn z A-1 B zn y , B zn B z , B . 0 , A0 A-1 X в Y0 , 0 , , 7.3.10 y = A-1 B z . 0 7.4.15. . X . 7.4.16. . , . . (. 2.3.15). 7.4.17. . . . ·
2

·

1

X .

146 (X, ·

. 7. IX ) I @ @ IX @ (X, · 6 X (X, ·
1

1

2

)

)

X . IX . 7.4.18. . X, Y T L (X, Y ) . x gr T := x X + T x Y x X . · gr T · X . , (X, · gr T ) . , · gr T · X . . 7.4.19. . X , X , - . 7.4.20. . X X = X1 X2 , X1 , X2 Lat(X ). X1 X2 , . : . : Z - , . . Y T B (Y , Z ) : Z = T (Y ). , , , , T . z 0 := T -1 z Y . , (Z, · 0 ) z = T T -1 z T T -1 z = T z 0 , . . · 0 · Z . X1 X2 , (X1 , · 1 ) (X2 , · 2 ). · k · X Xk k := 1, 2. x1 X1 x2 X2 x1 + x2 0 := x1 1 + x2 2 . X · , · X . (X, · 0 ) . 7.4.17.

7.5. 7.5.

147

7.5.1. . f : X R· () X . : (1) U := int dom f = f |U ; (2) V , sup f (V ) < +. (1) (2): . (2) (1): , U = . 7.1.1, , u U W , f , . . t := sup f (W ) < +. , , u := 0, f (u) := 0 W . f R+ , 1, v W : f (v ) = f (v + (1 - )0) f (v ) + (1 - )f (0) = f (v ); f (v ) + f (-v ) f (v ) + f ((-v )) = =2 1 1 f (v ) + f (-v ) 2 2 2f (0) = 0.

, |f (W )| t, f u := 0. 7.5.2. . x int dom f f x, x (f ) . l x (f ), ( x X ) l(x) l(x) + f (x) - f (x) , , l x. , l 7.5.1. 5.3.7, , l . 7.5.3. . . 7.5.4. . f : X R· , epi f .

148

. 7.

7.5.5. . . f . core dom f = , . x core dom f , t := f (x) F := (epi f )-1 R в X . , > 0 F (t + BR ) x + BX . , , f (x + BX ) t + 1. 7.5.1, f int dom f . x int dom f , , 7.1.1, core dom f = int dom f . 7.5.6. . 7.3.6, , f , , int dom f . , x0 int dom f , -, L > 0, -, U , f (x) - f (x0 ) L x - x0 , x U . 7.5.7. . f : X R· X x core dom f . f (x) x (f ) X . . 7.5.8. . 7.5.7 f : X R· x X , . . x (f ) := x (f ) X . () . ( ) , X # , ( ) X , , . 7.5.2 7.5.7 . , , , f := p X , : | |(p) := | |(p) X .

7.5.

149

7.5.9. . f : Y R· Y . , , X T B (X, Y ). x X , T x core dom f , x (f T ) = T x (f ) T . . l X x (f T ), 3.5.3 l1 f T x, l = l1 T . , , 7.5.7, l1 Y , , T x (f ). 7.5.10. . X, Y , T B (X, Y ) p : Y R . | |(p T ) = | |(p) T . l | | (p T ), l = l1 T l1 p (. 3.7.11). 7.5.2 , l1 . , | |(p T ) | | (p) T . . 7.5.11. . X0 X l0 X0 . l X , l0 . ( , l = l0 .) p := l0 · , : X0 X . 7.5.10 l0 | | (p ) = | | (p) = l0 | |( · ) . , | | ( · X ) = BX . 7.5.12. . U X . L X L int U = , H X , H L H int U = .

150

. 7.

7.5.13. . 7.5.12 , . 7.5.14. . X
0

X .

cl X0 = {ker f : f X , ker f X0 }. , (f X , ker f X0 ) ker f cl X0 . x0 cl X0 , x0 , / cl X0 . 7.5.12 7.5.13 f0 (XR ) , ker f0 cl X0 f0 (x0 ) = 1. , Re-1 f0 X0 x0 . , . 7.6. 7.6.1. X, Y () ( F) X , Y . , , T X Y . y Y y T X y y T . 7.6.2. . T : Y X , 7.6.1, T : X Y . 7.6.3. . T T B (X, Y ) B (Y , X ). , B (X, Y ) L (Y , X ), . , y = sup{|l(y ) : l | |( · )}, T = sup{ T y y : y 1} =

= sup{|y (T x)| : = sup{ T x : .

1, x 1} = x 1} = T ,

7.6.

151

7.6.4. . (1) X, Y , T B (X, Y ). , T B (X, Y ) T B (X , Y ). (·)X : X X X , . . x x := ( · , x) (·)Y : Y Y Y , . . y y := ( · , y ). T B (Y , X ) T B (Y , X ) : X -Y (·)X (·)Y X -Y , , y Y T y = (T y ) . x X T y (x) = y (T x) = (T x, y ) = (x, T y ) = (T y ) (x). x . (2) : X0 X X0 X . : X X0 , (x )(x0 ) = x (x0 ) x0 X0 x X , . . X - X0 0
T T T T

.

7.6.5. . X Y . Y - X . , , , . 7.6.6. . X - Y T , X - Y . X -Y X -Y
T T T

152

. 7.

: X X : Y Y X X Y Y (. 5.1.10 (8)). x X . , T x = (T x) . y Y . T x (y ) = x (T y ) = T y (x) = y (T x) = (T x) (y ). y Y . 7.6.7. . , . , X - Y
R S T

X - Y
R S

T

Z

Z

. R = S T R = (S T ) = T S , . , R = S T . 7.6.6, (Rx) = R x = S T x = S (T x ) = S (T x) = (S T x) x X . , R = S T . 7.6.8. . X X .
0

X , X

0

X0 := {f X : ker f X0 } = | |( (X0 ));

X0 := {x X : f X0 f (x) = 0} = {ker f : f X0 }.

X0 () X0 , X0 () X0 . .

7.6.9. . X, Y . T B (X, Y ) , im T .

7.6.

153

7.6.10. T B (X, Y ) , T , X im T , . : . : 7.4.2. 7.6.11. . T B (X, Y ). (1) (im T ) = ker(T ); (2) T , im T =

ker(T ),

(ker T ) = im(T ).

(1) y ker(T ) T y = 0 ( x X ) T y (x) = 0 ( x X ) y (T x) = 0 y (im T ) . (2) cl im T = ker(T ) 7.5.13. , im T . x = T y T x = 0, x (x) = T y (x) = y (T x) = 0, . . x (ker T ) . , im(T ) (ker T ) . x (ker T ) . , T im T , , X - im T x y0 F
T

- Y y

y 0 (im T ) , y 0 T = x . y Y , y y 0 . , x = T y , . . x im(T ). 7.6.12. . X, Y . T B (X, Y ) , T B (Y , X ). : 7.6.11 (2), im(T ) = (ker T ) . (ker T ) , , . : cl im T = Y . , 0 = Y = (cl im T ) = (im T ) = ker(T ) 7.6.11. S B (im(T ), Y ), S T = IY . r := S = 0 . , T y 1/r q y y Y .

154

. 7.

, cl T (BX ) 1/2r BY . , T (BX ) T (BX ) 1/4r BY . , T . y cl T (BX ). , y / , T (BX ). , , X Y , , F := R. 7.5.12 y Y , y y y (y ) sup y (T x) = T y
x 1

1 y. r

y 1/r > 1/2r. , T . . Y0 := cl im T , : Y0 Y . T = T , T : X Y0 , T x = T x x X . , im(T ) = im(T ) = T (im( )) = T (Y0 ), (Y ) = Y0 (. 7.6.4 (2)). , T . , T . , im T = im T . 7.6.13. . . . . - X
k-1
k - Xk - X

T

Tk

+1

k+1

- . . .

, . . . - X
k-1

k Xk - X -

T

Tk

+1

k+1

- . . . . .

: im Tk+1 = ker Tk+2 , T , ker(Tk ) = (im Tk ) = (ker T
k+1

k+1

) = im(T

k+1

).

155
k+1

: T 7.6.11 (2),

. ) .
k

(im Tk ) = ker(Tk ) = im(T

k+1

) = (ker T

k+1

Tk 7.6.12, im T . 7.5.14, im T k = ((im Tk ) ) = ((ker T , ker T
k+1 k+1

) ) = ker T

k+1

.

.

7.6.14. . T (ker T ) coker(T ) (coker T ) ker(T ). 2.3.5 (6) 0 ker T X Y coker T 0 . 7.6.13 , 0 (coker T ) Y X (ker T ) 0 . 7.6.15. . T T .
T T

7.6.16. . Sp(T ) = Sp(T ).
7.1. , . 7.2. , . 7.3. ? 7.4. X, Y T : X Y . , t R T x Y t x X , ker T = 0 im T . 7.5. .

156

. 7.

7.6. T X l1 (E). ker T . 7.7. , C ([a, b]), C (1) ([a, b]), . 7.8. X Y , X Y . , X Y . 7.9. X1 , X2 , X1 X2 = 0. , X1 + X2 , inf { x1 - x . 7.10. (amn ) , , (x(m) ) l1 , (m) ax (). , n=1 mn n x l1 , a x () n=1 mn n m N. 7.11. T H , T x | y = x | T y x, y H . T . 7.12. X + X : X = X + - X + . , t > 0 , x X x = x1 - x2 , x1 , x2 X + , : x1 t x , x2 t x . 7.13. p, q X , dom p dom q dom p - dom q = dom q - dom p X . , (. 3.10) (p + q ) = p + q . 7.14. p , X , T X . , T p. , p T . 7.15. f : X R· X f f (x ) := sup{ x | x - f (x) : x dom f }
2

/x

1

: x1 = 0, x1 X1 , x2 X2 }

(x X ); (x X ).

(x) := sup{ x | x - f (x ) : x dom(f )}

, f f = f

.

7.16. , l .

157

7.17. X , X . , c0 lp (1 p +) . 7.18. X Y T B (X, Y ) , im T . T . 7.19. X0 X , X0 X/X0 . X .

8

8.1. 8.1.1. . X . BX X ( X ), x X x = sup{|l(x)| : l }. , , U X , sup{|l(u)| : u U } < + l , sup U < +, . 8.1.2. . (1) BX 5.1.10 (8) 7.2.7. (2) 0 () 0 BX , 1 () . 1 (3) ext(BX ) 3.6.5 BX = | |( · X ), . (, , C ([0, 1], R)). (4) X, Y ( F) Y Y .
B

:= {

(y , x)

:y

Y

, x BX },

8.1.
(y , x)

159

(T ) := y (T x) y Y , x X T B (X, Y ). , |
(y , x)

(T )| = |y (T x)| y

Tx y

T

x,

. .

(y , x)

B (X, Y ) . , T B (X, Y ) x 1} = sup{|y (T x)| : y
(y , x) Y

T = sup{ T x :

, x 1} =

= sup{|

(T )| :

(y , x)

B

}.

, B ( B (X, Y )). Y , B . , U , {|y (T x)| : T U } x BX y Y , {T x : T U } Y x X . 7.2.5 , sup U < +. 8.1.3. . X X . , , f : D X D C X , D ( C R R2 ). : (1) z0 D lim f (z ) - f (z0 ) ; z - z0

z z0

(2) z0 D l lim

z z0

l f (z ) - l f (z0 ) , z - z0

. . l f : D C l . (1) (2): . (2) (1): , z0 = 0 f (z0 ) = 0. 2, D , . . 2D D , D := BC . , D () T,

160

. 8.

T ( ) T := {z C : |z | = 1}. z1 , z2 D \ 0 l f ( l ) : 1 l f (zk ) = zk 2 i
2T

l f (z ) dz z (z - zk )

(k := 1, 2).

, z1 = z2 , , z 2T |z - zk | (k := 1, 2), l f D , 1 f (z1 ) f (z2 ) l = - z1 - z2 z1 z2 1 1 · z1 - z2 2 i
2T

=

l f (z )

1 1 - z (z - z1 ) z (z - z2 )

dz =

=

1 2
2T

l f (z )

1 dz z (z - z1 )(z - z2 )

M sup |l f (z )| < +
z 2T

M > 0. , : sup
z1 =z2 ;z1 ,z2 =0 |z1 |,|z2 |

-

1 |z1 - z2 |

f (z1 ) f (z2 ) < +. - z1 z2

. 8.1.4. . f : D X , 8.1.3 (1) (, , 8.1.3 (2) - ), .

8.1.

161

8.1.5. . . , f 8.1.3 (1), f . , f 8.1.3 (2) := BX , f . 8.1.3 (2) 8.1.2 (4), . . f : D B (X, Y ), Y := BY := B , . , : . 8.1.6. . X - f (z )dz (. 4.8.5), . . , () : T ( ) f (z )dz :=
T

f d ,

, , (. 5.5.9 (6)). , . . . 8.1.7. . D f : D X X . , , F (, D ). f (z )dz = 0.
F

z0 int F f (z0 ) = 1 2 i
F

f (z ) dz . z - z0

162

. 8.

8.1.3 . , , , , 8.1.2 (1) 5.5.9 (6) . 8.1.8. . X - . 8.1.9. . f : D X z0 D . U := {z C : |z - z0 | < } , cl U D , ( )

f (z ) =
n=0

cn (z - z0 )n ,

cn cn = 1 2 i
U

f (z ) (z - z0 )n

+1

dz =

1 dn f (z0 ). n! dz n

u (u - z )-1 f (z ) = z - z0 , . . 1 1 = u-z (u - z0 ) 1 -
z -z0 u-z0

1 2 i
U

f (u) du u-z

(z cl U )

=

=
n=0

(z - z0 )n . (u - z0 )n+1

u U . ( U = U + q D - q > 0, cl U D .) ,

8.1.

163

sup f ( U ) < +, , f (z ) z cl U . U 8.1.7, , U . U , U . 8.1.10. . f : C X sup f (C) < +, f . > 0, D 8.1.9, cn sup f (z ) ·
z T -n

sup f (C) ·

-n

n N > 0. , cn = 0 n N. 8.1.11. . T . Sp(T ) = , R(T , · ) C, , 5.6.21. , 5.6.15, R(T , ) 0 || +. 8.1.10 , R(T , · ) = 0. , 5.6.15, , || > T R(T , )( - T ) = 1. . 8.1.12. Ё r(T ) = sup{|| : Sp(T )} T B (X ), X , . . . , r(T ) , 5.6.16. , r(T ) = 0 . r(T ) > 0. C , || > sup{|µ| : µ Sp(T )}. ||-1 (. 5.6.15) f (z ) := R (T , z -1 ), z = 0, z 0, z = 0.
-1

:

res(T ),

8.1.9 5.6.17, , ||-1 < r(T )-1 . , || > r(T ).

164

. 8.

8.1.13. K C H (K ) K , . . (f H (K ) f : dom f C , dom f K ). f1 , f2 H (K ) f1 f2 , D dom f1 dom f2 , K D f1 |D = f2 |D . H (K ). 8.1.14. . 8.1.13 H (K ) := H (K )/. f H (K ), f H (K ), f K . 8.1.15. f , g H (K ). f1 , f2 f , g1 , g2 g . , ,

x dom f1 dom g1 1 (x) := f1 (x) + g1 (x), x dom f2 dom g2 2 (x) := f2 (x) + g2 (x). 1 , 2 H (K ), 1 = 2 . K D1 dom f1 dom f2 K D2 dom g1 dom g2 , f1 f2 g1 g2 , , D1 D2 1 2 . 8.1.16. . , 8.1.15, f1 f2 f1 + f2 . . 8.1.17. H (K ) , 8.1.16, . 8.1.18. . H (K ) K . 8.1.19. K C, R : C\K X X . , , f H (K ) f1 , f2 f . F1 (K, dom f1 ), F2 (K, dom f2 ), f1 (z )R(z )dz =
F
1

f2 (z )R(z )dz .
F
2

8.2.

165

K D int F1 int F2 , D f1 |D = f2 |D . K F D . f1 R dom f1 \ K f2 R dom f2 \ K , f1 (z )R(z )dz =
F F
1

f1 (z )R(z )dz ,

f2 (z )R(z )dz =
F F
2

f2 (z )R(z )dz

( ). f1 f2 D . 8.1.20. . h H (K ), 8.1.19 h R h(z )R(z )dz :=
F

f (z )R(z )dz ,

h = f F

(K, dom f ).

8.1.21. . h(z ) 8.1.20 . , z K f1 , f2 h w := f1 (z ) = f2 (z ). w h z h(z ) = w. , 8.1.20 R U \ K , int U K . 8.2. 8.2.1. . X () T X , . . T B (X ). h H (Sp(T )) R(T , · ) T RT h := 1 2 i h(z )R(T , z )dz

166

. 8.

( h). f , Sp(T ), f (T ) := RT f := RT f . f (T ) = 1 2 i f (z ) dz . z-T

8.2.2. . , , . . . A1 , A2 ( ). A1 A2 A1 A2 ( A2 ) R, . . R L (A1 , A2 ) , R(ab) = R(a)R(b) a, b A1 . R , ker R = 0. R : A1 A2 A1 A2 . A2 () L (X ) X ( ), A1 A2 () A1 X A1 . X A1 . X R A X1 , R(a) a A, R1 : A L (X1 ), R1 (a)x1 = R(a)x1 x1 X1 a A, R ( X1 ). X = X1 X2 R(a) a A, , R () R1 R2 ( X1 X2 ). (= , ). 8.2.3. . RT T X n T . f (z ) = n=0 cn z

8.2.

167

( Sp(T )), f (T ) = n=0 cn T n ( B (X )). , RT , . RT . f1 , f2 H (Sp(T )) F1 , F2 , Sp(T ) int F1 F1 int F2 F2 D , f1 f1 , f2 f2 D . , 5.6.19, RT f1 RT f2 = f1 (T )f2 (T ) = 11 2 i 2 i
F
1

f1 (z1 ) dz1 z1 - T
F
2

f2 (z2 ) dz2 = z2 - T

= 11 2 i 2 i
F
2

f1 (z1 )R(T , z1 )dz1 f2 (z2 )R(T , z2 )dz2 =
F
1

=

11 2 i 2 i
F1 F
2

f1 (z1 )f2 (z2 )R(T , z1 )R(T , z2 )dz2 dz1 = R(T , z1 ) - R(T , z2 ) dz2 dz1 = z2 - z1 f2 (z2 ) dz2 R(T , z1 )dz1 - z2 - z1
F
2

=

11 2 i 2 i
F1 F
2

f1 (z1 )f2 (z2 )

1 = 2 i
F
1

1 f1 (z1 ) 2 i 1 f2 (z2 ) 2 i

1 - 2 i
F
2

f1 (z1 ) dz1 R(T , z2 )dz2 = z2 - z1
F
1

=

1 2 i

f1 (z1 )f2 (z1 )R(T , z1 )dz1 - 0 = f1 f2 (T ) = RT (f1 f2 ).

:= T, res(T ), f (z ) = n=0 cn z n . 5.6.16 5.5.9 (6),

168

. 8. 1 2 i

f (T ) =

f (z )
n=0

z

-n-1

T n dz =

=

1 2 i

f (z )z
n=0

-n-1

T n dz =

=
n=0

1 2 i

f (z ) n dz T = z n+1

cn T
n=0

n

8.1.9. 8.2.4. . 8.2.3 . 8.2.5. . f H (Sp(T )), T B (X ), f (Sp(T )) = Sp(f (T )). , Sp(f (T )) f -1 () z (C \ f -1 ()) dom f g (z ) := g Sp(T ), f ) = ( - f )g = 1C . 8.2.3, , . , f -1 () Sp Sp(f (T )) f (Sp(T )). Sp(T ). = z g (z ) := f () - f (z ) ; -z Sp(T ) = . ( - f (z ))-1 . g ( - res(f (T )). (T ) = , . .

g () := f ().

, g ( ). 8.2.3 g (T )( - T ) = ( - T )g (T ) = f () - f (T ). , f () res(f (T )), R(f (T ), f ())g (T ) - T . , res(T ), . , f () C \ res(f (T )) = Sp(f (T )), . . f (Sp(T )) Sp(f (T )).

8.2.

169

8.2.6. K ; g : dom g C , dom g K . f H (g (K )) g (f ) := f g . g H (g (K )) H (K ). 8.2.7. . g : dom g C, dom g Sp(T ) T B (X ), : g H (Sp(T )) H (Sp(g (T ))) @ @ Rg(T ) @ RT @ R @? B (X ) f H (g (Sp(T ))) f : D C , f f D g (Sp(T )) = Sp(g (T )). F1 (Sp(g (T )), D ) F2 (Sp(T ), g -1 (int F1 )). , g ( F2 ) int F1 , , z2 (z1 - g (z2 ))-1 int F2 z1 F1 . , 8.2.3 R(g (T ), z1 ) = 1 2 i
F
2

R(T , z2 ) dz z1 - g (z2 )

2

(z1 F1 ).

, Rg
(T )

f=

1 2 i
F
1

f (z1 ) dz1 = z1 - g (T ) R(T , z2 ) dz2 dz1 = z1 - g (z2 )
F
2

11 = 2 i 2 i
F
1

f (z1 )

f (z1 ) dz1 R(T , z2 )dz2 = z1 - g (z2 )
F
1

=

11 2 i 2 i
F
2

170 = 1 2 i
F
2

. 8. f (g (z2 ))R(T , z2 )dz2 = RT g (f )

( g (z2 ) int F1 z2 F2 , f (g (z2 )) = 1 2 i
F
1

f (z1 ) dz1 . z1 - g (z2 )

8.2.8. . : f g (T ) = f (g (T )) f H (g (Sp(T ))). 8.2.9. . Sp(T ) T , , := Sp (T ) \ . 8.2.10. () , . , , P := (T ) := 1 2 i (z ) dz . z-T

P X () X := im P T . 2 = , , 8.2.3, (T )2 = (T ). , T = RT IC , IC : z z , T P = P T ( IC = IC ). , 2.2.9 (4), X T . 8.2.11. . P 8.2.10 , . 8.2.12. . T B (X ). X X = X X , T T T 0 0 T ,

8.2. T T X T T X Sp(T ) = , Sp(T ) = .

171 ,

+ = Sp(T ) = 1C , 8.2.3 8.2.10 T . 8.2.5 8.2.3 0 = IC (Sp(T )) = Sp ( IC (T )) = Sp (RT ( IC )) = = Sp(RT RT IC ) = Sp(P T ). P T T 0 0 0 .

. - P T -T 0

0

,

. . - P T ( 0) \ 0 . , 0 Sp D z

, - T . , Sp(T ) \ 0 Sp(P T ) \ 0 = (T ) 0 . / D , D , 0 D D , / D h(z ) :=

zD

1 ; z h(z ) := 0.

8.2.3, h(T )T = T h(T ) = P . , h = h, X = X X h(T ) h(T ) h(T ) X h(T ) T = T h(T ) = 1. , T , . . 0 Sp(T ). , , / 0 . , Sp(T ) . , res(T ) = res(T ) res(T ). , Sp(T ) = C \ res(T ) = C \ (res(T ) res(T )) = = (C \ res(T )) (C \ res(T )) = Sp(T ) Sp(T ) = Sp(T ). , = , .

172

. 8.

8.2.13. . T B (X ). X = X X RT H (Sp(T )) X R R . : H (Sp(T )) @ @ R

-

H ( ) R
T

H (Sp(T )) @ @

H ( )

@

@ R @? B (X )

RT @ R @ R @? B (X )

(f ) := f , (f ) := f f H (Sp(T )) , f . 8.3. 8.3.1. X, Y . K L (X, Y ) : (1) K : K K (X, Y ); (2) U X V Y , K (U ) V ; (3) K X Y ; (4) X ( K ) Y ; (5) (xn )nN BX (K xn )nN . 8.3.2. . X, Y . (1) K (X, Y ) B (X, Y ); (2) W Z B (Y , Z ) K (X, Y ) B (W, X ) K (W, Z ),

8.3.

173

. . S B (W, X ), T B (Y , Z ), K K (X, Y ), T K S K (W, Z ); (3) IF K (F) := K (F, F) F. , K (X, Y ) B (X, Y ), 8.3.1. Kn K (X, Y ) Kn K , > 0 n K x - Kn x K - Kn x , x BX . , Kn (BX ) - (= B -) K (BX ). 4.6.4, K (X, Y ). . 8.3.3. . 8.3.2 : . , K (X ) := K (X, X ) ( ) B (X ), . . K (X ) B (X ) K (X ) B (X ) K (X ) K (X ). 8.3.4. . 0, K (l2 ), B (l2 ) B (l2 ) l2 . 8.3.5. . 8.3.4 , B (X )/K (X ), ( X ). 8.5. 8.3.6. . T L (X, Y ) , T B (X, Y ) im T . T F (X, Y ). 8.3.7. : T F (X, Y )
n

( x1 , . . . , xn X , y1 , . . . , yn Y ) T =
k=1

xk yk .

8.3.8. . Q T B (X, Y ) T
Q

() X .

:= sup T (Q) .

174

. 8.

· Q B (X, Y ) B (X, Y ) B (X,Y ) . . 8.3.9. . X . : (1) > 0 Q X T F (X ) := F (X, X ) , T x - x x Q; (2) W F (W, X ) B (W, X ) B (W,X ) ; (3) Y F (X, Y ) B (X, Y ) B (X, Y ) . , (2) (1) (3) (1). , (1) (2) (1) (3). (1) (2): T B (W, X ) = Q W W , , 4.4.5, T (Q) X , , > 0 T0 F (X ) , T0 - IX T (Q) = T0 T - T Q . , T0 T F (W, X ). (1) (3): T B (X, Y ). T = 0, . T = 0, > 0 Q X . T0 F (X ) , T0 - IX Q T -1 . T T0 - T Q T T0 - IX Q . , T T0 F (X, Y ). 8.3.10. . , ( , ) 8.3.9 (1)8.3.9 (3), . 8.3.11. . X , W cl F (W, X ) = K (W, X ), . 8.3.12. . (, , ), -

8.4.

175

. . 70- . . 8.3.13. . . B (l2 )

8.3.14. . lp p = 2 c0 , . 8.4.

8.4.1. -. X0 X X = X0 . > 0 X - X0 , . . x X , x = 1 d(x , X0 ) := inf d · ({x } в X0 ) 1 - . 1 > x X \ X0 . , d := d(x, X0 ) > 0. X0 x , x - x d/(1 - ) ( , d/(1 - ) > d). x := (x - x ) x - x -1 . x = 1. , x0 X0 x-x = x-x d(x, X0 ) ( x - x x0 + x ) - x 1 - . x -x x0 - x = x0 -

=

1 x -x

8.4.2. . X . X , X . . , X , X X1 X2 . . . , Xn+1 = Xn n N. 8.4.1 (xn ), xn+1 Xn+1 , xn+1 = 1 d(xn+1 , Xn ) 1/2, . . 1/2- Xn Xn+1 . , d(xm , xk ) 1/2 m = k . , (xn ) . , 8.3.1 IX .

176

. 8.

8.4.3. T K (X, Y ), X, Y . T , T . . Y0 := im T Y . 7.4.4 T (BX ) Y0 . , T T (BX ) Y0 . 8.4.2 Y0 . 8.4.4. X K K (X ). 1 - K . T := 1 - K . X1 := ker T . , X1 8.4.2. 7.4.11 (1) . X2 X1 . , X2 T (X ) = T (X2 ), , t > 0 T x t x x X2 . (xn ) , xn = 1, xn X2 T xn 0. K , , (K xn ) . y := lim K xn . (xn ) y , y = lim(T xn + K xn ) = lim xn . T y = lim T xn = 0, . . y X1 . , , y X2 . , y X1 X2 , . . y = 0. ( y = lim xn = 1). 8.4.5. > 0 . , (n )nN K , |n | n N. , , 0 = xn ker(n - K ) , n . , {xn : n N} . , , n {x1 , . . . , xn }. , xn+1 = k=1 k xk . n 0 = (n+1 - K )xn+1 = k=1 k (n+1 - k )xk . , k = 0 k := 1, . . . , n. xn+1 = 0.

8.4.

177

Xn := L ({x1 , . . . , xn }). X1 X2 . . . , , , Xn+1 = Xn n N. 8.4.1 (xn ) , xn+1 Xn+1 , xn+1 = 1 d(xn+1 , Xn ) 1/2. m > k , z := (m+1 - K )xm+1 Xm z + K xk Xm + Xk Xm . , K xm+1 - K xk = - =
m+1 xm+1 m+1 xm+1

+ K xm+1 +
m+1

m+1 xm+1

- K xk =

. 2 , (K xn ) . - (z + K xk ) | |d(xm+1 , Xm ) 8.4.6. . X, Y ( F). K K (X, Y ) K K (Y , X ). : , x x |BX X l (BX ). K (BY ) V := {K y |BX : y BY }. , x BX y BY K y |BX (x) = y K |BX (x) = y (K x), Q := cl K (BX ) K : C (Q, F) l (BX ), K g : x

g (K x). , K , , . S := {y |Q : y BY }. , S C (Q, F). , 4.6.10, S . 4.4.5 ,

K (S ). , y BY K y |Q = K y |BX , . . K (S ) = V . : K K (Y , X ), K K (X , Y ). 7.6.6, K |X = K . , K . 8.4.7. (. . ).

178

. 8.

8.4.4 7.6.13, , , . 8.4.5 8.4.6, , . 8.4.8. . , , . , . . K K (X ) 0 = Sp(K ) ker( - K ) = 0. F := C. , {} . g (z ) := 1/z g (z ) := 0 z {} , : {} = g IC . , 8.2.3 8.2.10, P{} = g (K )K . 8.3.2 (2), P{} K (X ). 8.4.3 , im P{} . 8.2.12. F := R . , X 2 C, i(x, y ) := (-y , x). X iX . X iX K (x, y ) := (K x, K y ). X iX (. 7.3.2), , K , Sp(K ). , K . , K . 8.4.9. . X , f : C C , , T B (X ) f (T ) K (X ). T P{} .

8.5. Ё

179

, . . (n )nN Sp(T ) , n = 0 ( , X ). f (n ) f (), f () = 0 . 8.2.5, Sp(f (T )) = f (Sp(T )). , 8.4.8 n f (n ) = f (). , f (z ) = f () z C , , f (T ) = f (). 8.4.2 X . , , Sp(T ). g (z ) := f (z )-1 , , g f = {} . , 8.2.3, P{} = g (T )f (T ), . . 8.3.2 (2) P{} . 8.4.10. . 8.4.9 . 8.5. Ё 8.5.1. . X, Y ( F). T B (X, Y ) Ё T N (X, Y ), ker T := T -1 (0) coker T := Y / im T , . . (T ) := dim ker T ; (T ) := dim coker T . ind T := (T ) - (T ) T . 8.5.2. . Ё . 8.5.3. Ё . 7.4.20. 8.5.4. T B (X, Y ) T N (X, Y ) T N (Y , X ). ind T = - ind T . 2.3.5 (6), 8.5.3, 5.5.4 7.6.13 : 0 ker T X Y coker T 0;
T

180

. 8. 0 (ker T ) X Y (coker T ) 0; 0 ker(T ) Y X coker(T ) 0; 0 (ker(T )) Y X (coker(T )) 0
T T T

. (T ) = (T ) (T ) = (T ) (. 7.6.14). 8.5.5. , . 8.5.4. 8.5.6. . T . (1) T x = 0 . T y = 0 . T x = y , , . T y = x , , . (2) T x = 0 . T y = 0 . T x = 0 x1 , . . . , xn . T y = 0 y 1 , . . . , y n . T x = y , y 1 (y ) = . . . = y n (y ) = 0. x x0 , . .
n

x = x0 +
k=1

k xk

(k F ).

8.5. Ё

181

T y = x , x (x1 ) = . . . = x (xn ) = 0. y y 0 , . .
n

y = y0 +
k=1

µk y

k

(µk F ).

8.5.5 7.6.11. 8.5.7. . (1) T , T . (2) T L (F n , F m ). rank T . (T ) = n - rank T ; (T ) = m - rank T . T N (F n , F m ) ind T = n - m. (3) X = X1 X2 T B (X ). X T1 0 T . 0 T2

T := dim im T , , T

, T Ё , Ё . (T ) = (T1 ) + (T2 ), (T ) = (T1 ) + (T2 ), . . ind T = ind T1 + ind T2 . 8.5.8. . K K (X ). 1 - K . , F := C. 1 / Sp(K ), 1 - K ind (1 - K ) = 0. 1 Sp(K ), 8.4.8 8.2.12 X = X1 X2 , X1 , 1 Sp(K2 ), K2 / K X2 , 1-K 1-K 0
1

0 1-K

.
2

8.5.7 (2), ind (1 - K1 ) = 0. 8.5.7 (3) ind (1 - K ) = ind (1 - K1 ) + ind (1 - K2 ) = 0.

182

. 8.

F := R , 8.4.8. , X iX K (x, y ) := (K x, K y ). ind (1 - K ) = 0. , R C (1 - K ) = (1 - K ) (1 - K ) = (1 - K ). ind (1 - K ) = 0. 8.5.9. . T B (X, Y ). L B (Y , X ) T , LT - 1 K (X ). R B (Y , X ) T , T R - 1 K (Y ). S B (Y , X ) T B (X, Y ), S T . T , T . 8.5.10. L R T . L - R K (Y , X ). LT = 1 + KX (KX K (X )) LT R = R + KX R; T R = 1 + KY (KY K (Y )) LT R = L + LKY 8.5.11. L T K K (Y , X ), L + K T . (L + K )T - 1 = (LT - 1) + K T K (X ) 8.5.12. , . K := L - R K T . . L, R T . 8.5.10, (Y , X ). , 8.5.11, R = L - K , R T .

8.5.13. . , X = Y S T , (S )(T ) = (T )(S ) = 1, : B (X ) B (X )/K (X ) . , . .

8.5. Ё

183

8.5.14. Ё . Ё , . : T N (X, Y ). 7.4.10, X = ker T X1 Y = im T Y1 P B (X ) ker T X1 Q B (Y ) Y1 im T . , T1 := T |X1 - T1 : X1 im T . S := T1 1 (1 - Q). S B (Y , X ). , S T + P = 1 T S + Q = 1. : S T , . . S T = 1 + KX T S = 1 + KY KX KY . , ker T ker(1 + KX ), . . ker T ker(1 + KX ), 8.5.8. , im T im(1 + KY ), . . - 1 + KY T . 8.5.15. . T N (X, Y ) S B (Y , X ) T , S N (Y , X ).

8.5.16. . Ё Ё . ( ) . 8.5.17. 0 X1 X2 . . . X
n-1

Xn 0

.
n

(-1)k dim Xk = 0.
k=1

n = 1 0 X1 = 0, n = 2 0 X1 X X1 X2 (. 2.3.5 (4)). n := 1, 2 . , m n - 1, n . 0 X1 X2 . . . X
n-2

X1 0 , 2 0 , > 2,
Tn-
1

-- X --

Tn-

2

n-1

- - Xn 0 --

184

. 8.

0 X1 X2 . . . X
n-2 n-2

- - ker T -

Tn-

2

n-1

0.

(-1)k dim Xk + (-1)n
k=1

-1

dim ker T

n-1

= 0.

, T dim X
n-1

n-1

,
n-1

= dim ker T

+ dim Xn .

n-2

0=
k=1

(-1)k dim Xk + (-1)n
n

-1

(dim X

n-1

- dim Xn ) =

=

(-1)k dim Xk .
k=1

8.5.18. . Ё . T N (X, Y ) S N (Y , Z ). 8.5.16, S T N (X, Z ). 2.3.16, 0 ker T ker S T ker S coker T coker S T coker S 0. 8.5.17 (T ) - (S T ) + (S ) - (T ) + (S T ) - (S ) = 0, ind (S T ) = ind S + ind T . 8.5.19. . T . ind T = - ind ind (S T ) = ind (1 + K . 8.5.8, 1 + T Ё S S. K ) K .

8.5. Ё 8.5.20. N (X, Y ) K K (X, Y ), T + K ind T . S KY K (Y ) S T = 1 + KX ;

185

. Ё : T N (X, Y ) ind (T + K ) = T , . . K
X

K (X )

TS = 1 + K

Y

( S 8.5.14). , S (T + K ) = S T + S K = 1 + KX + S K 1 + K (X ); (T + K )S = T S + K S = 1 + KY + K S 1 + K (Y ), . . S T + K . 8.5.14, T + K N (X, Y ). 8.5.19 ind (T + K ) = - ind S ind T = - ind S . 8.5.21. . Ё : N (X, Y ) , ind : N (X, Y ) Z . T N (X, Y ). 8.5.14 S B (Y , X ), KX K (X ) KY K (Y ) , S T = 1 + KX ; T S = 1 + KY .

S = 0, X Y 8.4.2, . . 8.5.7 (2). S = 0, V B (X, Y ), V < 1/ S , 5.6.1 : S V < 1 V S < 1. , 5.6.10 1 + S V 1 + V S B (X ) B (Y ) . (1 + S V )-1 S (T + V ) = (1 + S V )-1 (1 + KX + S V ) = = 1 + (1 + S V )-1 K
X

1 + K (X ),

186

. 8.

. . (1 + S V )-1 S T + V . , S (1 + V S )-1 T + V . , (T + V )S (1 + V S )-1 = (1 + KY + V S )(1 + V S )-1 = = 1 + KY (1 + V S )-1 1 + K (Y ). 8.5.12, T + V . 8.5.14, T + V N (X, Y ). N (X, Y ). , Ё (. 8.5.12), 8.5.19 8.5.18 ind (T + V ) = - ind ((1 + S V )-1 S ) = = - ind (1 + S V )-1 - ind S = - ind S = ind T ( (1 + S V )-1 8.5.7 (1)). . 8.5.22. . , . : T N (X, Y ) ind T = 0. X = X1 ker T Y = im T Y1 . , T1 T X1 X1 im T . , 8.5.5, dim Y1 = (T ) = (T ), . . Id : ker T Y1 . , T T T1 0 0 0 = T1 0 0 Id + 0 0 0 - Id .

: T := S + K , K K (X, Y ) S -1 B (Y , X ), , 8.5.20 8.5.7 (1), ind T = ind (S + K ) = ind S = 0. 8.5.23. . Inv(X, Y ) X Y ( 5.6.12). F (X, Y )

187

, X Y . : F (X, Y ) = Inv(X, Y ) + K (X, Y ). 8.5.22, , F (X, Y ) = Inv(X, Y ) + F (X, Y ), , , F (X, Y ) B (X, Y ).
8.1. . .

8.2.

8.3. (fn ) , U T . , (fn ) U (fn (T )) . 8.4. T . , := Sp(T ) \ a r , {z C : |z - a| < r}. , P P = lim (1 - z -n (T - a)n )-1 ;
n

x im(P ) lim sup (a - T )n x
n

1 n

< r.

8.5. , . 8.6. , , , . 8.7. , , ( ). 8.8. 1 p < r < +. , lr lp c0 lp .

188

. 8.

8.9. H . T B (H ) (en )
1/2

T

2

:=
n=1

T en

2

.

( !) . , T , 2 < +, (n ) n=1 n (T T )1/2 . 8.10. T . im(T 0 ) im(T 1 ) im(T 2 ) . . . . n , im(T n ) = im(T n+1 ), , T . n T d(T ). ker(T 0 ) ker(T 1 ) ker(T 2 ) . . . a(T ). , T a(T ) d(T ) . 8.11. T , T Ё . , T , T = U + V , U , V ( ) U. 8.12. T X r := a(T ) = d(T ). , im(T r ) ker(T r ) , X = ker(T r ) im(T r ) T T im(T r ) . 8.13. T . (T ) := dim ker T , (T ) := dim coker T ,

T ( Ё ).
+

(X ) := {T B (X ) : im T Cl(X ), (T ) < +}; (X ) := {T B (X ) : im T Cl(X ), (T ) < +}. T T
+ -

-

, (X ) T (X ) T
- +

(X ); ( X ).

8.14. T . , T + (X ) , , U T (U ) X .

189

8.15. T , ( - T ) Ё . , T , C, = 0 : () ( - T ) ; () ( - T )k k N; () ( - T )k k N, , , T , T (= ). 8.16. : (X/ Y ) Y Y X Y , Y X .

X /Y

8.17. , T f H (Sp(T )) f (T ) . 8.18. , , ( ). 8.19. A, B Ё B (X, Y ). ind A = ind B , , A B B (X, Y ).

9

9.1. 9.1.1. . X . : X P (P (X )) X , (1) x X (x) X ; (2) x X (x) fil{x}. (x) ( ) x. (X, ) ( X ) . 9.1.2. . T (X ) X . 1 , 2 T (X ), , 1 2 ( 1 2 ) : x X 1 (x) 2 (x). 9.1.3. T (X ) . X = , T (X ) = {} . X = , 1.3.13. 9.1.4. . G X , () (: G Op( ) ( x G)(G (x))). F X , (: F Cl( ) X \ F Op( )). 9.1.5. . -

9.1.

191

. 9.1.6. (X, ) x X , .

U t( )(x) ( V Op( )) x V & U V . t( ) : x t( )(x) X .

9.1.7. . X , = t( ). (X, ) ( X ) . X T(X ). 9.1.8. . (1) . (2) . (3) := inf T (X ). , (x) = {X } x X . , Op( ) = {, X } , , = t( ), . . . . (4) := sup T (X ). , (x) = fil {x} x X . , Op( ) = 2X , , = t( ), . . . . (5) Op X , . , , X , Op( ) = Op. (x) := fil {V Op : x V } x X ( X = ). , (x) = , X (.: inf = +). , t( ) = Op Op( ). G Op( ), G = {V : V Op, V G} , , G Op . . 9.1.9. t : T (X ) T (X ) t : t( ). (1) im t = T(X ), . . T (X ) t( ) T(X );

192

. 9.

(2) 1 2 t(1 ) t(2 ) (1 , 2 T (X )); (3) t t = t; (4) T (X ) t( ) ; (5) Op( ) = Op(t( )) ( T (X )). Op( ) Op(t( )) , . Op( ) Op(t( )) t( ). Op( ) = Op(t( )) . 9.1.10. X , x X ( U (x))( V (x) & V U ) ( y )(y V V (y )). 9.1.9 (5). 9.1.11. 1 , 2 T(X ). : (1) 1 2 ; (2) Op(1 ) Op(2 ); (3) Cl(1 ) Cl(2 ). 9.1.12. . 9.1.8 (5) 9.1.11, . Op( ) X . , (X, ) X (X, t( )) . t( ) , . 9.1.13. . T(X ) X . E T(X ) sup t(supT
(X ) T(X )

E = sup

T (X )

E. E t(sup E ).

E ) sup

T (X )

t(E ) sup

T (X )

T (X )

, := supT (X ) E T(X ). , E . , 0 E 0 T(X ), 0 , , = supT(X ) E . 1.2.14.

9.2.

193

9.1.14. . : inf T(X ) E = t(inf T (X ) E ). , 9.1.12 , : U Op(inf , Op(inf
T(X ) T(X )

E ) ( E ) U Op( ).

E) =
E

Op( ).

( ). 9.2. 9.2.1. . , , , , . . . , 4.1.19 4.2.1. 9.2.2. . : (1) ; (2) , ; (3) , . 9.2.3. f : (1) ; (2) ; (3) x f (x);

194

. 9. (4) x , x, f , f (x); (5) , x, f , f (x).

9.2.4. . , , ( , ) 9.2.3 (1)9.2.3 (5), . 9.2.5. . f : (X, X ) (Y , Y ) 9.2.3 (5) x X , , f x (. 4.2.2). , . , x (x) := X (x) x (x) := fil {x} x X , x = x, f x ( X X ) f : (X, x ) (Y , Y ) ( X x ). 9.2.6. 1 , 2 T(X ). 1 2 , IX : (X, 1 ) (X, 2 ) . 9.2.7. f : (X, ) (Y , ) 1 T(X ) 1 T(Y ) , 1 1 . f : (X, 1 ) (Y , 1 ) . (X, ) IX (X, 1 ) - -
f f

(Y , ) IY (Y , 1 )

, . 9.2.8. . f : X (Y , ). T0 := { T(X ) : f : (X, ) (Y , ) }. f
-1

( ) := inf T0 T0 .

9.2. 9.2.3 (1) T0 (x X f
-1

195

( (f (x))) (x)).

(x) := f -1 ( (f (x))). , t( ) = . , f ( (x)) = f (f -1 ( (f (x)))) (f (x)), . . T0 9.2.3 (3). , : f -1 ( ) = . 9.2.9. . f f .
-1

( )

9.2.10. . 9.2.8 : , . , , , 9.1.14, . , (x x f -1 ( )) (f (x ) f (x) ); (F x f -1 ( )) (f (F ) f (x) ) F . 9.2.11. 0 := { T(Y f ( ) := sup 0 9.1.13 f ( )(y ) = (sup
T(Y ) 0

. f : (X, ) Y . ) : f : (X, ) (Y , ) }. 0 . y Y
T (Y ) 0

)(y ) = (sup

)(y ) = sup{ (y ) :

0

}.

9.2.3 (3)
0

(x X f ( (x)) (f (x))).
0

, , f ( )

.

9.2.12. . f ( ) f . 9.2.13. . 9.2.11 : , . 9.2.14. . (f : X (Y , )) - . , , := sup f 1 ( ). (= ) X , f ( ).

196 9.2.8,

. 9.

- (f : (X, ) (Y , ) ) f 1 ( ).

9.2.15. . (f : (X , ) Y ) . , , := inf f ( ). (= ) Y , f ( ). 9.2.11, : (f : (X , ) (Y , ) ) f ( ). 9.2.16. . 9.2.14 9.2.15 . 9.2.17. . (1) (X, ) X0 X . : X0 X X0 X . 0 := -1 ( ). 0 ( X0 ), (X0 , 0 ) (X, ). (2) (X , ) . , , X := X (X ) . := sup Pr-1 ( ), Pr : X X , Pr x = x ( ). , ( ) , . (X, ) . , X := [0, 1] , X := [0, 1] ( ) . := N . 9.3. 9.3.1. : (1) ; (2) ; (3) , .

9.3. , y cl{x} ( V (y )) x V x {V : V (y )}, x, y (X, ).

197

9.3.2. . , ( , ) 9.3.1 (1)9.3.1 (3), T1 -. T1 - ( T1 -, ). 9.3.3. . : . T1 -

9.3.4. : (1) ; (2) ; (3) . (1) (2): y U (x) cl U , V (y ) , U V = , U (x). , F := (x) (y ). , F x F y . x = y . (2) (3): x, y X , x = y ( , X = , X ). U (x) , U = cl U y = U . , V U X . , U V = . (3) (1): F X . F x F y , F (x) F (y ). , U (x) V (y ) U V = . , x = y . 9.3.5. . , ( ) 9.3.4 (1)9.3.4 (3), T2 -. .

198

. 9.

9.3.6. . : T2 - , . 9.3.7. . U , V . , V U , int V U. 9.3.8. : (1) ; (2) , ; (3) , , . (1) (2): x X U (x), V := X \ int U x V . F Cl( ), / x F int F V . G := X \ F . , G (x). / G X \ int F = cl(X \ int F ) X \ V int U U . , cl G U . (2) (3): x X F Cl( ), x F , X \ F (x). / , U = cl U (x), X \ F . , X \ U F , U. (3) (1): F Cl( ) int G F y cl G, U (y ) G F U G = . , y F . 9.3.9. . , ( , ) 9.3.8 (1)9.3.8 (3), T3 -. T3 - . 9.3.10. . : (1) , ;

9.3.

199

(2) . (1) (2): F1 , F2 X , F1 F2 = . G := X \ F1 . , G G F2 . F2 = , . , , F2 = . V2 , G V2 int V2 F2 . V1 := X \ V2 . , V1 , V1 V2 = . V1 X \ G = X \ (X \ F1 ) = F1 . (2) (1): F = cl F , G = int G G F . F1 := X \ G. F1 = cl F1 , , U U1 , U U1 = , F U F1 U1 . , cl U X \ U1 X \ F1 = G. 9.3.11. . , ( ) 9.3.10 (1), 9.3.10 (2), T4 -. T4 . 9.3.12. , . T R t Ut (t T ) X . , , f : X R , {f < t} Ut {f t} (t T ) , t, s T , t < s cl Ut int Us . : t < s {f t} {f < s} cl Ut {f t} {f < s} int Us . : Ut cl Ut int Us Us t < s, t Ut (t T ) . f 3.8.2 ( 3.8.4). t Vt := cl Ut t Wt := int Ut . . , 3.8.2, g , h : X R , t T {g < t} Vt {g t}, {h < t} Wt {h t}.

200

. 9.

t, s T , t < s, 3.8.3 Wt = int Ut Ut Us f h; Vt = cl Ut int Us = Ws h g ; Ut Us cl Us = Vs g f . f = g = h. 3.8.4 9.1.5, t R {f < t} = {h < t} = {Ws : s < t, s T } Op(X ); {f t} = {g t} = {Vs : t < s, s T } Cl(X ). f . 9.3.13. . X T4 -. , , F X G . f : X [0, 1] , f (x) = 0 x F f (x) = 1 x G. / Ut := t < 0 Ut := X t > 1. Ut T - [0, 1] , . . T := nN Tn , Tn := {k 2-n+1 : k := 0, 1, . . . , 2n-1 }, , t Ut (t T := T (R \ [0, 1])) 9.3.12. . t T1 , . . t {0, 1}, U0 := F , U1 := G. , t Tn n 1 Ut , cl Ut int Us , t, s Tn t < s. t Tn+1 t Tn , . . tl := sup{s Tn : s t}; tr := inf {s Tn : t s}. t = tl t = tr , Ut . t = tl t = tr , tl < t < tr cl Utl int Utr . 9.3.11 Ut , cl Utl int Ut Ut = cl Ut int Utr . , .

9.4.

201

, t, s Tn+1 , t < s. tr = sl , s > sl cl Ut cl Utr = cl Usl int Us . t < tr = sl cl Ut int Utr = inf Usl int Us . tr < sl , , , cl Ut cl Utr int Usl int Us , . 9.3.14. . X T4 - , F1 , F2 X , f : X [0, 1] , f (x) = 0 x F1 f (x) = 1 x F2 . : 9.3.13 F := F1 G := X \ F2 . : F1 F2 = F1 , F2 , G1 := {f < 1/2} G2 := {f > 1/2} f ; G1 F1 , G2 F2 . 9.3.15. . X 1 T3 2 -, x X F , x, f : X [0, 1] , f (x) = 1 y F f (y ) = 0. 1 T3 2 - . 9.3.16. . 9.3.1 9.3.14. 9.4. 9.4.1. B cl B := {cl B : B B } . (1) cl B = cl fil B ; (2) B x x cl B ; (3) (B , x cl B ) B x.

202

. 9.

(3), (1) (2) . U (x) B B U B = . , F := (x) B . , F x. , F = B , B . 9.4.2. . , (. 4.4.1). 9.4.3. . X C X . : (1) C ; (2) B C , B B , B C = ; (3) , C , C ; (4) , C , C . (1) (2): cl B C = , C X \ cl B . , C X \ {cl B : B B } = {X \ cl B : B B }. , B0 B , C {X \ cl B0 : B0 B0 } = X \ {cl B0 : B0 B0 }. B B , B {B0 : B0 B0 } {cl B0 : B0 B0 }. C B C ({cl B0 : B0 B0 }) = . (2) (3): C = , . C = , B B B C = , C B . , cl B C = . (3) (4): 9.4.1. (4) (1): , C = ( ). , C . E , C {G : G E },

9.4.

203

E0 E , C {G : G E0 }. B :=
GE0

X \ G : E0

E

.

, B

. ,

cl B = {cl B : B B } = {X \ G : G E } = = X \ {G : G E } X \ C. F , B ( 1.3.10). B C , , C F . F x x C , , 9.4.1 (2), cl F C = . cl F cl B . . 9.4.4. . (1) (4) 9.4.3 X = C : , (. 4.4.7). , . : . . , , . 9.4.5. . (. 4.4.5). 9.4.6. X0 X C X0 . C X0 , C X . : 9.4.5 9.2.17 (1). : B X0 . , , V := clX0 B B , X0 . , V C = . B X , W := clX B , X . , V = W X0 , , W C = . - C X 9.4.3 B B , B C = . 9.4.3, , C X0 .

204

. 9.

9.4.7. . 9.4.6 : , . . , . , . . , . 9.4.8. . . X := X . X , X = . X = F X. 1.3.12 Pr : X X , Pr (F ) X . , 9.4.3 x X , Pr (F ) x . x : x . , F x (. 9.2.10). 9.4.3, , X . 9.4.9. . X C Cl(X ). , , F X C F . 9.4.3 X : F x. 9.2.2, x cl C = C . 9.4.3, , C . 9.4.10. . C X . C = , . C = x cl C . 9.2.2 F0 , C F0 F0 x. F , F0 . F x C F . 9.4.3 F C . 9.3.4 . , x C . 9.4.11. f : (X, ) (Y , ) , f (X ) = Y . , , f . , f -1 . , F Cl( ) f (F ) Cl( ). F Cl( ). F 9.4.9. 9.4.5 9.4.10, , f (F ) .

9.4.

205

9.4.12. 1 2 X . (X, 1 ) , (X, 2 ) 1 2 , 1 = 2 . 9.4.13. . 9.4.12 . 9.4.14. . . X B - X . , , U cl B . , X \ int U (. 9.4.9), cl B (X \ int U ) = . 9.4.3 B B , B (X \ int U ) = , . . B U . , , B := {cl B : B B }, , cl B U . x X B := (x). 9.3.4, cl B = {x} , , (x) , . , X . F X . B F . 9.3.8, cl B = F , B , . 9.3.9, X . 9.4.15. . . , , 9.4.8 9.4.9. , . X . Q X [0, 1]. : X [0, 1]Q (x)(f ) := f (x), x X f Q. 9.4.14 9.3.14 , X (X ). , . 9.4.11. 9.4.16. . 9.4.15 . , ( ) .

206

. 9.

9.4.17. . , , (. 4.5 4.6). 9.4.18. . F , G1 , . . . , Gn , F G1 . . . Gn . F1 , . . . , Fn , F = F1 . . . Fn Fk Gk (k := 1, . . . , n). n := 2. k := 1, 2 Uk := F \ Gk U1 U2 = . 9.3.10 V1 V2 , U1 V1 , U2 V2 V1 V2 = . Fk := F \Vk . , Fk Fk F \Uk = F \(F \Gk ) Gk k := 1, 2. F1 F2 = F \ (V1 V2 ) = F . 9.4.19. . 9.3.14 , 9.4.18 X n h1 , . . . , hn : X [0, 1] , hk |Gk = 0 k=1 hk (x) = 1 x F . ( , Gk := X \ Gk .) 9.4.20. . , , . , . 9.4.21. , (= ), . . . : 9.4.5 , ( ) . 9.4.9 9.4.6. : X X · := X {}, X . X · X . X X · X . A X · K X , A K , K A. A K X , A .

9.5.

207

9.4.22. . X , X · , 9.4.20, X . 9.5. 9.5.1. . X UX X 2 . UX X , (1) UX fil {IX }; (2) U UX U -1 UX ; (3) ( U UX )( V UX ) V V U . X UX := {}. (X, UX ) ( X ) . 9.5.2. (X, UX ) x X (x) := {U (x) : U UX }. : x (x) X . , , . W (x), W = U (x) U UX . V UX , V V U . y V (x), V (y ) V (V (x)) = V V (x) U (x) W . , W y y V (x). , V (x) int W . , int W x. 9.1.6. 9.5.3. . , 9.5.2, (X, UX ) (UX ), X . . 9.5.4. . (X, ) , U X , (U ). 9.5.5. . (1) ( ) ( ).

208

. 9.

(2) ( ) ( ). (3) f : X (Y , UY ) f -1 (UY ) := f в-1 (UY ), , , f в (x1 , x2 ) := (f (x1 ), f (x2 )) (x1 , x2 ) X 2 . , f -1 (UY ) X . (f
-1

(UY )) = f

-1

( (UY )).

f -1 (UY ) UY f . , . (4) (X , U ) . , , X := X -1 . UX := sup Pr (U ). UX . , (UX ) (X , (U )) . (5) , . 9.4.15 X . 9.5.5 (3) 9.5.5 (4) X . , , , IX , UX . , . (6) X , Y , UY Y B 2X . B B UY UB , := {(f , g ) Y
X

вY

X

: g IB f

-1

}.

U := fil {UB , : B B , UY } Y X , ( ) : B . , , (. 8.3.8). , B X , U Y X .

9.5.

209

, ( ). B {X }, U , (U ) Y X . 9.5.6. . , ( ) , , , . . , , 4.2.44.2.9, 4.5.8, 4.5.9, 4.6.14.6.7. , , . . 9.5.7. . X , R· := {x R· : x 0}. d : X 2 R+ X , (1) d(x, x) = 0 (x X ); (2) d(x, y ) = d(y , x) (x, y X ); (3) d(x, y ) d(x, z ) + d(z , y ) (x, y , z X ). (X, d) . 9.5.8. (X, d) Ud := fil {{d } : > 0}. Ud .
+

9.5.9. . M () X . (X, M) , M . UM := sup{Ud : d M}. 9.5.10. . , . .

210

. 9.

9.5.11. X , Y , Z , T R (Ut )tT , (Vt )tT , X в Z Z в Y . , , f : X в Z R, , {f < t} Ut {f t}, {g < t} Vt {g t}, {h < t} Ut Vt {h t} (t T ). h(x, y ) = inf {f (x, z ) g (z , y ) : z Z }. 3.8.2. 3.8.4. h f g . 9.5.12. . f : X в Z R, g : Z в Y R. h, 9.5.11, - f g f

g : Z в Y R,

h:X вY R

g (x, y ) := inf {f (x, z ) g (z , y ) : z Z }.

+- f g f
+

g (x, y ) := inf {f (x, z ) + g (z , y ) : z Z }.

9.5.13. . f : X 2 R· + K - (K R, K 1), (1) f (x, x) = 0 (x X ); (2) f (x, y ) = f (y , x) (x, y X ); 1 (3) K f (x, u) f (x, y ) f (y , z ) f (z , u) (x, y , z , u X ). 9.5.14. . 9.5.13 (3) () . 9.5.12 K -1 f f f f .

9.5.

211

9.5.15. 2-. 2- f : X 2 R· d , 1/2f d + f. f1 := f ; fn+1 := fn + f (n N). fn
+1

(x, y ) fn (x, y ) + f (y , y ) = fn (x, y )

(x, y X ).

, (fn )

.
nN

d(x, y ) := lim fn (x, y ) = inf fn (x, y ). n N d(x, y ) f2n (x, y ) = fn
+ fn

(x, y ) fn (x, z ) + fn (z , y ),

d(x, y ) d(x, z ) + d(z , y ). 9.5.7 (1) 9.5.7 (2) . , 1/2 f d. , fn 1/2 f n N. n := 1, 2 . , f f1 . . . fn 1/2 f fn+1 (x, y ) < 1/2 f (x, y ) (x, y ) X 2 n 2. z1 , . . . , zn X t := f (x, z1 ) + f (z1 , z2 ) + . . . + f (z 1 +f (zn , y ) < f (x, y ). 2
n-1

, zn )+

f (x, z1 ) t/2, t/2 f (z1 , z2 ) + . . . + f (zn , y ) 1/2 f (z1 , y ). , t f (x, z1 ) t f (z1 , y ). 9.5.13 (3), 1/2 f (x, y ) f (x, z1 ) f (z1 , y ) t. : 1/2 f (x, y ) > t 1/2 f (x, y ). , f (x, z1 ) < t/2. m N, m < n, f (x, z1 ) + . . . + f (z
m-1

, zm ) <

t ; 2 t . 2

f (x, z1 ) + . . . + f (zm , zm+1 )

212

. 9.

, m = n f (zn , y ) t/2. ( , t/2 f (x, z1 ) + . . . + f (zn-1 , zn ) 1/2 f (x, zn ) 1/2 f (x, y ) > t f (x, z n ) f (zn , y ) 1/2 f (x, y ).) f (z
m+1

,z

m+2

) + . . . + f (z

n-1

, zn ) + f (zn , y ) <

t . 2

, : f (x, zm ) 2(f (x, z1 ) + . . . + f (z f (zm , zm+1 ) t; f (z
m+1 m-1

, zm )) t;

, y ) 2(f (z

m+1

, zm+2 ) + . . . + f (zn , y )) t.

, 2- 1 f (x, y ) f (x, zm ) f (zm , zm+1 ) f (z 2
m+1

, y) t <

1 f (x, y ). 2

, . 9.5.16. . . (X, UX ) . V UX . V1 := V V -1 . -1 Vn UX , V = V , V UX , V V V Vn . Vn+1 := V . Vn Vn+1 Vn+1 Vn+1 Vn+1 IX IX Vn+1 , (Vn )nN . t R Ut , IX , Ut := Vinf { V1 , 2 X, t t ,0 t t < = < = > 0, 0, t < 1, 1, 1.

nN : t2

-n

}

9.6.

213

t Ut (t R) . f : X 2 R, (. 3.8.2, 3.8.4) {f < t} Ut {f t} (t R). Wt := U2t t R, s < t Us Us Us Wt . , 3.8.3 9.2.1 f 2-. 9.5.15, dV , 1/2 f dV f . , UdV = fil {Vn : n N}. , M := {dV : V UX } UM = UX . 9.5.17. . , T3 1 -. 2 9.5.18. . . 9.6. 9.6.1. . E , F U X , . . E , F 2X U (E ) (F ). , E F E F , E F , . . ( E E ) ( F F ) E F . 9.6.2. . E X ( X ), X ( ), E . . , X , . 9.6.3. e. E X . {GE : E E }, cl GE E E E .

214

. 9.

S s : E Op(X ), s(E ) = X E E s(E ) = E cl s(E ) E . s1 , s2 : s1 s2 := ( E E ) (s1 (E ) = E s2 (E ) = s1 (E )). , (S, ) , IE S . S . S0 S s0 (E ) := {s(E ) : s S0 } (E E ). s0 (E ) = E , s(E ) = E s S0 . s0 (E ) = E , s0 (E ) = {s(E ) : s(E ) = E , s S0 }. S0 : s0 (E ) = s(E ) s S0 , s(E ) = E . s0 (E ) Op(X ) s0 S0 . , s0 X (, , s0 S ). x X E1 , . . . , En E , x E1 . . . En x E E E . s0 (Ek ) = Ek / - k , x s0 (E ). , k s0 (Ek ) = Ek , s1 , . . . , sn S0 sk (Ek ) = Ek (k := 1, 2, . . . , n). S0 , , sn {s1 , . . . , sn-1 }. x sn (E ) E E E . , E {E1 , . . . , En } ( x E E ). / s0 (E ) = sn (E ), x s0 (E ). 1.2.20 S Ї Ї s. E E . F := X \ s(E \ {E }), F s(E ) Ї F . 9.3.10 G Ї Ї Op(X ) F G cl G s(E ). s(E ) := G s(E ) := s(E ) Ї Ї E = E (E E ). , s S . s(E ) = E , s s , , s = s. s(E ) cl G s(E ) = E , . . cl s(E ) E . Ї Ї Ї Ї s(E ) = E , cl s(E ) E . , s Ї Ї Ї . 9.6.4. . f (= ) X , . . f : X F. supp(f ) := cl{x X : f (x) = 0} f . supp(f ) , f . spt (f ) := supp(f ). 9.6.5. (fe )eE Ї X E := {supp(fe ) : e E } . Ї U , (fe )eE E Ї . E , (fe )eE , eE fe .

9.6.

215

, U (fe )eE . 9.6.6. . (f : X [0, 1])f F U X , U , f F f (x) = 1 x U . . . 9.6.7. . E U , F U . {supp(f ) : f F } E , F , E . F E : E . 9.6.8. . 9.6.3 {U : } {V : }, cl V U . 9.3.14 g : X [0, 1] , g (x) = 1 x V g (x) = 0 x X \ U . , supp(g ) U . 9.6.5 (g ) g . g (x) > 0 x X . f := g /g ( ). (f ) . 9.6.9. . , . 9.6.10. . . 9.6.11. . . 9.6.12. . , .

216

. 9.

9.6.13. . RN , (= ) (. 4.8.1). 9.6.14. . RN a () , a(x) > 0 |x| < 1 a(x) = 0 |x| 1. supp(a) = {x RN : |x| 1} B := BRN . 9.6.15. . () (b )>0 , , -, lim (sup | supp(b )|) = 0 , -, RN b (x) dx = 1
0

( > 0). - . . 9.6.16. . a(x) := t exp(-(|x|2 - 1)-1 ), int B, t RN a(x) dx = 1. a (x) := -N a(x/) (x RN ). 9.6.17. . f L1,loc (RN ), . . f (= ) . g Ё f g f g (x) :=
R
N

f (x - y )g (y ) dy

(x RN ).

9.6.18. . (a )>0 f (f a )>0 f L1,loc (RN ) (. 10.10.7 (5)). 9.6.19. : (1) K RN - U

9.6.

217

(= ) := K,U , . . : RN [0, 1], K int{ = 1} supp( ) U ; (2) U1 , . . . , Un Op(RN ), U1 . . . Un K . 1 , . . . , n : RN [0, 1], n supp(k ) Uk k=1 k (x) = 1 x K . (1) := d(K, RN \U ) := inf {|x-y | : x K, y U }. , / > 0. > 0 K + B. (b ) >0 := b . , + , := sup | supp(b )|, . (2) 9.4.18 Fk Uk , K . Kk := Fk K n k := Kk ,Uk . k / k=1 k (k := 1, . . . , n), n { k=1 k > 0}, n { k=1 k = 0} K . 9.6.20. RN := E . , RN E RN . E .

E A , ( := int )A . (V )A cl V A. 9.6.19 (1) := cl V , . (x) := (x)/ A (x) x (x) := 0 x RN \ , . 9.6.21. . , ( )A , K , , A0 A U K , A0 (x) = 1 x U (. 9.3.17, 9.6.19 (2)).

218

. 9.

9.1. , . 9.2. , ? 9.3. . 9.4. , . ? 9.5. (f : X (Y , )) . X ( ), (Z, ) g : Z X : g : (Z, ) (X, ) , f g ( ). , X, f ( ), ( ) . 9.6. (f : (X , ) Y ) . Y ( ), (Z, ) g : Y Z : g : (Y , ) (Z, ) , g f ( ). , Y , f ( ), ( ) . 9.7. , , , :

cl

A

=

cl A .

9.8. , , . 9.9. . 9.10. X H -, X X . , H - . 9.11. . 9.12. , .

219

9.13. , (?) . 9.14. A , B , A B = . , V V (A) V (B ) = . 9.15. , ( ) . 9.16. , . , , . 9.17. ? 9.18. , . 9.19. , . ?

10

10.1. 10.1.1. . (X, F, +, ·) F. X , , , : + : (X в X, в ) (X, ), · : (F в X, F в ) (X, ). (X, ) . 10.1.2.
X

. x x (x0 X, F \ 0)

x x + x0 ,

(X, X ). 10.1.3. . , X : (x + y ) = (x) + (y ) (, F \ 0; x, y X ),

(. 1.3.5 (1)) Ux+ y (x) + (y ) ( Ux (x) & Uy (y )) Ux + Uy Ux+ y .

10.1.

221

, . , , . , , . 10.1.4. . X N X . X , N = (0), , (1) N + N = N ; (2) N ; (3) N . (x) = x + N x X . : N = (0). 10.1.2 , (x) = x + N x X . , (1) ( X 2 ). (2) F (0)x N x X , . . x ( R) x X . (3) (2), , F (0)N = N , . . ( F в X ). : N , (1)(3). , N fil {0}. (x) := x + N . . (1) , , , . , X 2 . (2) (3) , (, x) x . x - 0 x0 = 0 (x - x0 ) + ( - 0 )x0 + ( - 0 )(x - x0 ) . , , N N F.

222

. 10.

n N, || n. V N W N , W W1 + . . . + Wn V , Wk := W . W = n (/n W ) nW W1 + . . . + Wn V . 10.1.5. . VT(X ) X . E VT(X ) sup
VT(X )

E = sup

T(X )

E.

:= supT(X ) E . E x x + x0 (X, ) (X, ), (X, ) (X, ). 9.1.13, , (0) 10.1.4 (1)10.1.4 (3), (0) E . 1.2.14. 10.1.6. . . T L (X, Y ) VT(Y ). := T -1 ( ). x x y y (X, ), , 9.2.8, T x T x, T y T y , , T (x + y ) T (x + y ). 9.2.10 , x + y x + y (X, ). , (x) = x + (0) x X , , (0) + (0) = (0). T -1 3.4.10 3.1.8, , (0) = T -1 ( (0)) , 10.1.4 (0). 10.1.4, : VT(X ). 10.1.7. . 10.1.5 10.1.6. 10.1.8. . A, B . , A B -, A + B A. -

10.2.

223

10.1.9. X , , U , IX , = (U ). U (0) VU := {(x, y ) X 2 : y - x U }. : IX VU ; VU + IX = VU ; (VU )-1 = V-U ; VU1 U2 VU1 VU2 ; VU1 VU2 VU1 +U2 U , U1 , U2 (0). 10.1.4, , U := fil {VU : U (0)} , = (U ). , U IX - . U , (U ) = , W IX - U , W = VW (0) . . 10.1.10. . (X, ) . U , 10.1.9, X . 10.1.11. . . 10.2. 10.2.1. . , , . 10.2.2. . X N X . X , N = (0), , 1 (1) 2 N = N ; (2) N , .

224

. 10.

: 10.1.2 x 2x . , 1/2 N = N . U N . V N , V U . 10.1.4, W , W V . 3.1.13 3.1.14, , co(W ) . W co(W ) V U . : . , N 10.1.4 (2), 10.1.4 (3). V N W , W N W V , 1/2 W N . , 1/2 W + 1/2 W W V - W . , N + N = N . 10.1.4. 10.2.3. . LCT (X ) X . E LCT (X ) sup
LCT (X )

E = sup

T(X )

E.

10.2.4. . . 10.2.5. . .

10.2.6. . 10.2.7. . X . X ( ) M . (X, M ) (X, ). 10.2.8. . . X := (M ) (X, M ). V (0). 10.2.2 B (0) , B V . 3.8.7 {pB < 1} B {pB 1}.

10.2.

225

, pB (. 7.5.1), . . pB M , , {pB < 1} (0). , V (0). , 5.2.10, (x) = x + (0) x + (0) = (x), . . . , . 10.2.9. . , , . 10.2.10. . 10.2.8 : . , (. 5.2.13). 10.2.11. . X . (X, ) (, , X ) X # , . (X, ) ( -) (X, ). 10.2.12. (X, ) = {| |(p) : p M }. 10.2.13. . (X, ) , LCT (X ) Lat(X # ), , . . E LCT (X ) (X, sup E ) = sup{(X, ) : E }. E = , sup E 0 X , , (X, 0 ) = 0 = inf Lat(X # ) = supLat(X # ) . 9.2.7 . 2.1.5, E (X, sup E ) sup{(X, ) : E }. f (X, sup E ) , 10.2.12 9.1.13 1 , . . . , n E , f (X, 1 . . . n ) . 10.2.12 5.3.7 p1 M1 , . . . , pn Mn , f | |(p1 . . . pn ). 3.5.7 3.7.9, , | |(p1 + . . . + pn ) = | |(p1 ) + . . . + | |(pn ). f (X, 1 ) + . . . + (X, n ) = (X, 1 ) . . . (X, n ) .

226

. 10. 10.3.

10.3.1. . X , Y F. , , (, , ) · | · X в Y F, . . , . x X y Y x| : y x | y , |y : x x | y , · | : X FY , | · : Y FX, X | Y #; |Y X # .

· | | · -. X | -, | Y -. 10.3.2. - - . -

10.3.3. . X Y , - - . , X Y , , Y X . ., X Y . - . 10.3.4. . (1) X Y · | · . (y , x) Y в X y | x := x | y . , Y X . . , , (. 10.3.3). , , Y X , X Y . , x | y R := Re x | y XR YR . , XR YR , . . x | y := x | y R , , x y .

10.3.

227

(2) H . H H . -. (3) (X, ) X . (x, x ) x (x) X X . (4) X , , X # := L (X, F) . , (x, x# ) x# (x) . 10.3.5. . X Y . X FY - - X , Y , (X, Y ). - (X, Y ) Y X - X Y Y , X . 10.3.6. - , -. - , -. x x ( (X, Y )) x x | ( FY ) ( y Y ) x | (y ) x | (y ) ( y Y ) x | y x | y ( y Y ) | y (x ) | y (x) ( y Y ) x x ( | y -1 (F )) 10.3.7. . (X, Y ), , 5.1.10 (4). (X, Y ) {| · | y | : y Y }. (Y , X ) {| x | · | : x X }. 10.3.8. (X, (X, Y )) (Y , (Y , X )) . 10.2.4 10.2.5. 10.3.9. . . X Y . 0 X -(Y , (Y , X )) 0, 0 Y -(X, (X, Y )) 0.
|· ·|

228

. 10. 10.3.6 -

- X Y Y X , . - 10.3.3. , 10.2.13 , (Y , (Y , X )) = (Y , sup{ x |-1 (F ) : x X }) = = sup{(Y , = L ({(Y , f
-1 -1

x |-1 (F )) : x X } = (F )) : f X |}) = X |,

5.3.7 2.3.12 (Y , f (F )) = {f : F} (f Y # ).

10.3.10. . 10.3.9 . , . , 10.3.9 10.3.4 (3) , , . 5.1.11 ( ) (x, y ) := x | y , . . , X Y X (Y , (Y , X )) , Y (X, (X, Y )) , X = Y Y = X . 10.4. , 10.4.1. . X Y X . , , (X, ) = | Y . , Y (X Y , Y X , . .) (Y , ) = X |. 10.4.2. . 10.3.9.

10.4. ,

229

10.4.3. (X, Y ) X , . (X, Y ) . E . 10.2.13 (X, (X, Y )) = (X, sup E ) = = sup{(X, ) : E } = sup{| Y : E } = | Y , E 10.4.2. 10.4.4. . (X, Y ), 10.4.3, . . X , X Y , ( X , X Y ). 10.4.5. . X X Y , (X, Y ) (X, Y ). 10.2.13 (X, ) , , . , , , 10.4.2 10.4.3 | Y = (X, (X, Y )) (X, ) (X, (X, Y )) = | Y . . 10.4.6. . , , . . 10.4.5 , U X (= ), U . p , p(U ) R. X0 := X/ ker p p0 := pX/ ker p . 5.2.14, , p0 . : X X0 . , (U ) (X0 , p0 ). 7.2.7 , (U ) p0 . p0 = p, U (X, p).

230

. 10.

10.4.7. . X . (X, X ) , X . 5.4.5, (X, X ), , , 5.3.4. 10.4.8. . (X, ) , K V X , K , V K V = . f (X, ) , sup Re f (K ) < inf Re f (V ). , , . K , W U := K + W V ( , K + W V + W , W ). 3.1.10 , U . , K int U = core U . 3.8.14 l (XR )# , , {l = 1} XR V U U . , l W , , l (XR , ) 7.5.1. f := Re-1 l, , 3.7.5, f (X, ) . , f . 10.4.9. . . . , 10.4.5 , U , U . , , 10.4.8, U {Re f t}, f () , t R. 10.5. 10.5.1. . X , Y F XвY . U X V Y -

10.5. (U ) := F (U ) := {y Y : F -1 (y ) U }; - -1 (V ) := F 1 (V ) := {x X : F (u) V }. (U ) () U , () V . 10.5.2. : (1) (u) := ({u}) = F (u), (U ) = uU (u); (2) ( U ) = (U ); - (3) F 1 (V ) = F -1 (V ); (4) U1 U2 (U1 ) (U2 ); (5) U в V F V (U ), U -1 (V ); (6) U -1 ( (U )).
-1

231

(V )

10.5.3. . U X Y , x X \ U y Y , U
-1

(y ),

x /

-1

(y ).

: U = -1 (V ), U = vV -1 (v ) 10.5.2 (1). : U -1 (y ) , y (U ). , U = y(U ) -1 (y ) = -1 ( (U )). 10.5.4. . -1 ( (U )) ( ) , U .
- 10.5.5. . F 1 (F (U )) U ( F ).

10.5.6. . (1) (X, ) , U X . (U ) U (. 1.2.7). (2) (H, (· , ·)H ) F := {(x, y ) H 2 : (x, y )H = 0}. U H (U ) = -1 (U ) = U . U U .

232

. 10.

(3) X X . F := {(x, x ) : x (x) = 0}. (X0 ) = X0 -1 (X0 ) = X0 X0 X X0 X (. 7.6.8). -1 ( (X0 )) = cl X0 7.5.14. 10.5.7. . X Y . pol := {(x, y ) X в Y : Re x | y 1}; abs pol := {(x, y ) X в Y : | x | y | 1}. pol (U ) (V ); abs pol U V ( U X V Y ). 10.5.8. . 2 (U ) := ( (U )) , U . 10.4.8 . 10.5.9. . U := (U ) , U . , , 10.5.8. 10.6. 10.6.1. X p : X R X . () (p) (X , X ). Q := xX [-p(-x), p(x)] Q . , (p) Q Q (p) , (X , X ). , (p) Q - p. 9.4.8 9.4.9, , (p) (X , X )- .

10.6.

233

10.6.2. . 10.6.3. . X . U X # ( ) sU : X R , U , (X # , X )-. : U = (sU ) sU . sU 3.6.6. 10.2.12 , (X, X # ) X (. 5.1.10 (2)). , sU (X, X # ). 10.6.1 U (X # , X ). U . : sU (x) := sup{l(x) : l U }. , sU dom sU = X . U (sU ). l (sU ) l U , / 10.4.8 10.3.9 x X sU (x) < l(x). . 10.6.4. . sU , 10.6.3, U . 10.6.5. . . U X . , X U = . 9.4.12, U (X, X ). (X, X ) # # X (X , X ) X , U = (sU ). (. 10.6.3) sU : X R sU (x ) := sup x (U ). 3.6.5 ext(U ) . ext(U ) 10.6.3. , sU , , U (. 3.6.6). 10.6.6. X Y S X . , , pS S . (S ) - ()

234 (pS ), . .

. 10. (S ) = | (pS )
-1 R

.

S , S - () | |(pS ), . . S = | | |(pS )
-1

.

y YR , y | (pS ) -1 , | y R R (pS ). , x S Re x | y = x | y R = | y R (x) pS (x) 1, S {pS 1} 3.8.7. , y (S ). , , y (S ), | y R (pS ). , x XR > pS (x) 1 > pS (-1 x), . . -1 x {pS < 1} S . -1 x | y R = Re -1 x | y = -1 Re x | y 1. | y R (x) . - , | y R (x) pS (x). , y | (ps ) -1 . R (S ) = | (pS ) -1 . R 3.7.3 3.7.9. 10.6.7. . . U X (U ) U ( X X ). U {p 1} p, 10.5.2 (4), (U ) ({p 1}) = (Bp ) = Bp . 10.6.6 , p Bp , , (U ) | |(p). 10.6.2 | |(p) (X , X )-. (U ) . 9.4.9, (X , X )- (U ). (U ) . 10.7. 10.7.1. . , .

10.7.

235

: X , . . (X ) = X . , X X . BX BX X X , BX (X , X )- 10.6.7. , BX ( ) BX , (X, X ) ( ) (X , X ). : X X . BX BX ( , (BX ) ). 10.5.9 , (X, X ) X (X , X ), , BX = BX (- , ). , X . 10.7.2. . , . 10.7.3. . . 10.4.9 , . , . 10.7.4. . ( ) . X , (X , X ) (X , X ), , 10.6.7, , BX (X , X )- . , X . X , X . X , , X . , X 10.7.3. 10.7.5. . , () .

236

. 10. 10.8. C (Q, R)

10.8.1. . Q (= ), C (Q, R) Q. C (Q, R) , · := · , Ё (. 4.6.8). : C (Q, R) , C (Q, R) . C (Q, R) - , . 10.8.2. . L C (Q, R) , f1 , f2 L f1 f2 L, f1 f2 L, , , f1 f2 (q ) := f1 (q ) f2 (q ), f1 f2 (q ) := f1 (q ) f2 (q ) (q Q).

10.8.3. . , C (Q, R) , , C (Q, R). 10.8.4. . (1) ; C (Q, R); . (2) . (3) L Q. Q0

-

LQ0 := {f C (Q, R) : ( g L) g (q ) = f (q ) (q Q0 )}. LQ0 . L LQ0 .

(4) Q0 Q. L C (Q, R) L
Q0

:= f

Q0

: f L .

10.8. C (Q, R) , LQ0 = f C (Q, R) : f
Q0

237

L

Q0

.

C (Q0 , R). L , L Q0 C (Q, R), . . C (Q, R), L Q 0 C (Q0 , R) (, Q0 = ). (5) Q := {1, 2}. C (Q, R) R2 . 2 R {(x1 , x2 ) R2 : 1 x1 = 2 x2 } 1 , 2 R+ . (6) L C (Q, R). q Q, : L{q} = C (Q, R), L{q} = {f C (Q, R) : f (q ) = 0}. q1 , q2 Q L {q ,q } = 0, 10.8.4 (5) 1 , 2 R+ 12 , L{
q1 ,q2 }

= {f C (Q, R) : 1 f (q1 ) = 2 f (q2 )}.

10.8.5. L C (Q, R). f C (Q, R) L , > 0 (x, y ) Q2 f := fx,y, L, f (x) - f (x) < , f (y ) - f (y ) > -.

: . : 3.2.10 3.2.11 , f = 0. > 0. x Q gy := fx,y, L. Vy := {q Q : gy (q ) > -}. Vy y Vy . Q y1 , . . . , yn Q, Q = Vy1 . . . Vyn . fx := gy1 . . . gyn . , fx L. , fx (x) < fx (y ) > - y Q. Ux := {q Q : fx (q ) < }. Ux x Ux . Q, x1 , . . . , xm Q , Q = Ux1 . . . Uxm . , , l := fx1 . . . fxm . , l L l < .

238

. 10.

10.8.6. . 10.8.5 (. 7.2.10). 10.8.7. . L C (Q, R) cl L = cl L{q1 ,q2 } .
(q1 ,q2 )Q2

cl L cl L{q1 . f cl L{q1 ,q2 10.8.5, f cl L.

,q2 } }

(q1 , q2 ) Q2 q1 , q2 , ,

10.8.8. . L C (Q, R) cl L =
(q1 ,q2 )Q2

L{

q1 ,q2 }

.

L{

q1 ,q2 }

.

10.8.9. . , U F Q Q, q1 , q2 Q , q1 = q2 , u U , : u(q1 ) = u(q2 ). 10.8.10. . , C (Q, R) C (Q, R). L , L{
q1 ,q2 }

= C (Q, R){

q1 ,q2 }

(q1 , q2 ) Q2 (. 10.8.4 (6)). 10.8.8. 10.8.11. µ C (Q, R) . N (µ) := {f C (Q, R) : [0, |f |] ker µ}. , , supp(µ) Q , f N (µ) f
supp(µ)

= 0.

10.8. C (Q, R) 3.2.15 [0, |f |] + [0, |g |] = [0, |f | + |g |].

239

, f , g N (µ) |f | + |g | N (µ). N (µ) , . . (f N (µ) & 0 |g | |f | g N (µ)), , N (µ) . , N (µ) . , fn 0, fn f fn N (µ). g [0, f ] g fn g g fn [0, fn ]. , µ(g ) = 0, . . f N (µ). 10.8.8, , N (µ) , N (µ) =
q Q

N (µ){q} .

supp(µ) : q supp(µ) N (µ){
q}

= C (Q, R) (f N (µ) f (q ) = 0). . -

, supp(µ) N (µ) =

N (µ){
q supp(µ)

q}

=

= {f C (Q, R) : f

supp(µ)

= 0}.

Q (. 9.4.14) 9.3.14. 10.8.12. . supp(µ), 10.8.11, µ (. 10.9.4 (5)). 10.8.13. . µ , N (µ) = {f C (Q, R) : µ(|f |) = 0}. , µ(f g ) = 0 g C (Q, R), f supp(µ) = 0. supp(µ) = N (µ) = C (Q, R)

240

. 10.

µ = 0. , . F Q. , F Ё µ X \ F µ, f , supp(f ) Q \ F , µ(|f |) = 0. supp(µ) Ё µ, µ Q supp(µ). , µ , µ (. 10.10.5 (6)). , 3.2.14 3.2.15 µ ( ) µ+ , µ- , |µ|, f C (Q, R)+ : µ+ (f ) = sup µ[0, f ]; µ- (f ) = - inf µ[0, f ]; |µ| = µ+ + µ- .

, C (Q, R) K - (. 3.2.16). 10.8.14. µ |µ| . N (µ) = N (|µ|). 10.8.15. 0 a 1 aµ : f µ(af ) f C (Q, R) µ C (Q, R) . |aµ| = a|µ|. f C (Q, R)+ (aµ)+ (f ) = sup{µ(ag ) : 0 g f } sup µ[0, af ] = = µ+ (af ) = aµ+ (f ). , µ+ = (aµ + (1 - a)µ)+ (aµ)+ + ((1 - a)µ)+ aµ+ + (1 - a)µ+ = µ+ . , (aµ)+ = aµ+ , . 10.8.16. . A C (Q, R) µ ext(A BC (Q, R) ). A µ . µ = 0, supp(µ) = . µ = 0, , , µ = 1. a A. A , , 0 a 1 q supp(µ) 0 < a(q ) < 1.

10.8. C (Q, R)

241

µ1 := aµ µ2 := (1 - a)µ. , µ1 + µ2 = µ, µ1 µ2 . , µ µ1 + µ2 = = sup µ(af ) + sup µ((1 - a)g ) =
f 1 g 1

sup
f 1, g 1

µ(af + (1 - a)g ) µ ,

aB
C (Q, R)

+ (1 - a)B

C (Q, R)

B

C (Q, R)

.

, µ = µ1 + µ2 . , µ = µ1 µ1 + µ2 µ1 µ2 , µ2

, µ1 , µ2 A , : µ1 = µ1 µ. 10.8.15, a|µ| = |aµ| = |µ1 | = µ1 |µ|. , |µ|((a - µ1 1)g ) = 0 g C (Q, R). 10.8.13 10.8.14, , a µ. 10.8.17. . C (Q, R) C (Q, R). 10.5.9 , A C (Q, R), A ( A ) C (Q, R) . 10.6.7, , A BC (Q,R) , 10.6.5 µ. , µ . µ , A Q. µ , µ . , supp(µ) . (. 10.8.13), µ . , , A C (Q, R).

242

. 10.

10.8.18. . C (Q, R) C (Q, R). pn , t [-1, 1] |pn (t) - |t| | 1 . 2n

|pn (0)| 1/2n. pn (t) := pn (t) - pn (0) |pn (t) - |t| | 1/n -1 t 1. pn . a A C (Q, R) a 1, |pn (a(q )) - |a(q )| | 1 n (q Q).

q pn (a(q )), , A. 10.8.19. . 10.8.18 ( 10.8.8) C (Q, R). , , 10.8.18 , - , t |t| [-1, 1]. 10.8.17, 10.8.18, . 10.8.20. . Q0 Q f0 C (Q0 , R). f C (Q, R) , f Q0 = f0 . Q0 = ( ). : Q0 Q : C (Q, R) C (Q0 , R), f := f . , . , im , C (Q0 , R), 10.8.17 (, , ) , im .

10.9.

243

coim := C (Q, R)/ ker . f C (Q, R) g := (f sup |f (Q0 )|1) (- sup |f (Q0 )|1). f
Q0

=g

Q0

, . . f := (f ) = (g ). , g

f . , f = inf = inf inf h h h
C (Q,R)

: (h - f ) = 0 =
Q0

C (Q,R)

:h

=f

Q0 Q0

=

Q0 C (Q,R)

:h

Q0

=f

= sup |f (Q0 )| = g f . , f = g = g = g
C (Q0 ,R) C (Q0 ,R)

=

= sup |g (Q0 )| = g = f ,

. . . 5.5.4 4.5.15, , coim , im C (Q0 , R). , im = im . 10.9. 10.9.1. . . K ( ) := K ( , F) := {f C ( , F) : supp(f ) }. Q , K (Q) := K (Q) := {f K ( ) : supp(f ) Q}. K (Q) · . E Op ( ) K (E ) := {K (Q) : Q E }. ( Q E E , Q Q E , .) 10.9.2. : (1) Q f C (Q, F) f
Q

=0

g K (Q) g

Q

=f

.

244

. 10. K (Q) ; Q, Q1 , Q2 Q Q1 в Q2 . C (Q, F) Q u1 · u2 (q1 , q2 ) := u1 u2 (q1 , q2 ) := u1 (q1 )u2 (q2 ) us K (Qs ) C (Q, F); , K ( ) = C ( , F). . C ( · , F), · := {} , K ( ) {f C ( · , F) : f () = 0}; E Op ( ) K (E ) Lat (K ( )) ; E , E Op ( ) :

(2)

(3)

(4) (5)

0 K (E E ) - - - K (E ) в K (E ) - - - K (E E ) 0, - - --
(E ,E )

(E

,E

)

(E ,E

)

f := (f , -f ),

(E ,E )

(f , g ) := f + g .

(1) Q int( \ Q). (2) . 9.3.13 10.8.17 (. 11.8.2). (3) , F = R. , K ( ) , 10.8.8 ( K ( ) · ) (. 10.8.11). (4) , K (sup ) = K () = 0. E Op ( ) E , f K (E ) : supp(f ) E E E ( supp(f )). K (E ) = {K (E ) : E E }. , , E1 , . . . , En Op ( ) f K (E1 . . . En ). 9.4.18 k K (Ek ) n n , k=1 k = 1. f = k=1 k f supp(f k ) Ek (k := 1, . . . , n). (5) (4). 10.9.3. . µ K ( , F)# ( , F-) µ M ( ) := M ( , F), µ K (Q) K (Q) , Q .

10.9. f dµ := f dµ := f (x) dµ(x) := µ(f ) (f K ( )).

245

µ(f ) f µ. µ . 10.9.4. . (1) q q : f f (q ) (f K ( )) . q - q . , q q -1 (s, t) в st , . . . e , e . () , . K ( ) a ( ) (a f )(q ) := a f (q ) := f (a-1 q ), (a f )(q ) := fa (q ) := f (q a-1 ) (f K ( ), q ) ( f в F). , a , a L (K ( )). (, ) M ( , R). () . () () () ( ). () . . RN . , . , f (a-1 x) dx = f (x) dx (f K ( ), a ).

(2) M ( ) := (K ( ), · ) . M ( ) . , C ( · , F) (. 10.9.2 (2)).

246

. 10.

(3) µ M ( ) µ (f ) = µ(f ) , f (q ) := f (q ) q f K ( ). µ µ. µ µ F = C. µ = µ , C-. , µ = µ1 + iµ2 , µ1 , µ2 C. , C- R- ( M ( , R)), K ( , C) K ( , R) iK ( , R). R-, , K -. () ( ) (. 5.5.9 (4), 5.5.9 (5)). µ |µ|, f K ( , R), f 0, |µ|(f ) := sup{|µ(g )| : g K ( , F), |g | f }. , . µ , |µ| | | = 0. µ, , µ. = f µ, f L1,loc (µ) f µ ( f µ) (f µ)(g ) := µ(f g ) (g K ( )) (= ). (4) Op ( ) µ M ( ), µ := µ K ( ) . µ µ M ( ) M ( ) : µ M( ) µ = (µ ) . : M : E Op ( ) M (E ) (= M ) ( ). , . (5) E Op ( ) µ M ( ). , E µ \ E Ё µ, µE = 0. 10.9.2 (4) supp(µ), µ,

10.9.

247

µ. , supp(µ) = supp(|µ|). 10.8.12. q {q }. (6) k µk M ( k ) (k := 1, 2). 1 в 2 , , µ , uk K ( k ) u1 (x)u2 (y ) dµ(x, y ) =
1

u1 (x) dµ1 (x)
1 2

u2 (y ) dµ2 (y ).

в

2

µ1 в µ2 := µ1 µ2 := µ. 10.9.2 (4), , f K ( 1 в 2 ) µ1 в µ2 (f ) (= ). (7) G _ µ, M (G). f K (G) f (s, t) := f (st) _ | µ f . µ |(µ в )(f ) _ (f ) := (µв )(f ) (f K (G)), Ё µ . , : µ =
GвG

s t dµ(s)d (t) = µ t d (t).
G

=
G

s dµ(s) =

Ё Ё M (G). , G . L1 (G), m, Ё ( M (G)). G. , f , g L1 (G) Ё (. 9.6.17): (f g )dm = f dm g dm. Ё µ M (G) f L1 (G) (µ f )dm := µ (f dm), . . Ё . , , f g =
G

x g f (x) dm(x) =
G

x (g )f (x) dm(x).

248

. 10.

. T B (L1 (G)). : (i) µ M (G) , T f = µ f f L1 (G); (ii) T : T a = a T a G, a K (G) L1 (G); (iii) T (f g ) = (T f ) g f , g L1 (G); (iv) T (f ) = (T f ) M (G), f L1 (G). 10.9.5. . K ( ) M ( ) ( K ( ) K ( )# ). M ( ) (M ( ), K ( )), . K ( ) K ( ) := (K ( ), M ( )) (, , (K ( ), K ( ) ) = M ( )). M ( ) , , : µ := sup{|µ(f )| : f 1, f K ( )} (µ M ( )). 10.9.6. K ( ) , K (Q) K ( ) Q, Q (. . K ( ) (. 9.2.15)). µ (K ( ), ) , µ M ( ), µ K (Q) Q . , µ M ( ) VQ := {f K (Q) : |µ(f )| 1} K (Q). , , {VQ : Q } = {f K ( ) : |µ(f )| 1} . , µ (K ( ), ) . K ( ) . , p , p M ( ). , q := p K (Q) K (Q) . 7.2.2 (- K (Q)) q . , K (Q) : K (Q) (K ( ), K ( ) ) K ( ) .

10.9.

249

10.9.7. A K (RN ) ( ), sup A < + , , A . Q RN , A K (Q). , n N qn RN an A, an (qn ) = 0 |qn | > n. B := {n|an (qn )|-1 qn : n N}, , , , p(f ) := sup{|µ|(|f |) : µ B } . p(an ) n|an (qn )|-1 qn (|an |) = n, A. 10.9.8. . (fn ) K (RN ). fn K 0, ( Q RN )( n) supp(fn ) Q & fn 0. 10.9.7 , µ K (RN )# , µ(fn ) 0, fn K 0. , , , . . . 10.9.9. . R (pn ) , pn dx n +. , Pn RN , Pn dx (, , dx := dx1 в . . . в dxN RN ). f K (RN ) f C (m) Q (. . ). Ё (f Pn ), , , Q f , m . RN ( 10.10.2 (4)). 10.9.10. . E (µE )E E : µE M (E ), (E , E ) E µE µE E E . , , µ , E µE E E.

250

. 10. 10.9.2 (5), K (E E ) -
{E ,E } E ,E E =E ,E E E

K (E ) - K ( ) 0,

(E ,E ) , . (. 10.9.6). . K ( ) = Q K (Q), 10.9.2 (4), . , n {E1 , . . . , En } (n 2) ,
n
n Kn -

n K (Ek ) - K (E1 . . . En ) 0,

k=1

n

Kn , Kn :=
k
n

K (Ek El ).

im n = ker n . n+1 (f , fn+1 ) = 0, f := (f1 , . . . , fn ), n f = -fn+1 fn+1 K ((E1 . . . En ) En+1 ). n , 10.9.2 (5), k K (Ek En+1 ) , := (1 , . . . , n ) n = -fn+1 . (f - ) ker n Kn , n = f - . ,
n

K

n+1

= Kn в
k=1

K (Ek E

n+1

)

( ), := ( , 1 , . . . , n ) Kn+1 n+1 = (f , fn+1 ). (. 7.6.13), 0 M ( ) -
E E

M (E ) -
{E ,E } E ,E E ,E =E

M (E E ).

.

10.10. D ( ) D ( )

251

10.9.11. . , , . 10.9.10 : M( ) , , M (. 10.9.4 (4)). 10.10. D () D () 10.10.1. . f D (RN ) := D (RN , F). Q f D (RN ) : supp(f ) Q : RN F. f RN Op (RN ) D (Q) := D ( ) := {D (Q) : Q }.

10.10.2. : (1) D (Q) = 0 int Q = ; (2) Q RN f :=
(Z+) 1 +...+N n
N

n,Q

:=
||n

f sup |(
1

C (Q)

:= f )(Q)|

...

n

( Q) f ( , Z+ := N {0}). MQ := { · n,Q : n N} D (Q) ; (3) C ( ) := E ( ) Op (RN ) M := { · n,Q : n N, Q } . D ( ) C ( ); (4) Q1 RN , Q2 RM Q Q1 в Q2 . D (Q) Q f1 f2 (q1 , q2 ) := f1 f2 (q1 , q2 ) := f1 (q1 )f2 (q2 ), qk Qk , fk D (Qk ), D (Q); (5) E Op ( ) D (E ) Lat (D ( )) : D (E E ) = D (E ) D (E ), D (E E ) = D (E ) + D (E ); D (E ) = L ( {D (E ) : E E }) (E Op ( )).

252

. 10. (. 10.9.2 (5)):
(E

- - - - 0 D (E E ) - - - D (E ) в D (E ) - - - D (E E ) 0.
,E ) (E ,E )

(1) (2) . (3) (Qm )mN , Qm , Qm Qm+1 , mN Qm = . { · n,Qm : n N, m N} M . 5.4.2 . . D ( ) C ( ) Tr ( ) := { D ( ) : 0 1}. Tr ( ) , 1 2 supp(1 ) int{2 = 1}. , f C ( ) ( f )Tr ( ) f . (4) a(q , q ) := a (q )a (q ), a , a RN RM , q RN q RM . f D (Q), m N > 0 f - f a m,Q /2. F := { f (q )q (a ) : || m, q Q1 в Q2 }, Q1 , Q2 , F 1/2 (N + 1)-m , в . f D (Q) , . . f - f m,Q . (5) 10.9.2 (4) 9.4.18 9.6.19 (2). 10.10.3. . 10.10.2 (4) , , . 10.10.4. . u D ( , F)# ( F) u D ( ) := D ( , F), u |D (Q) D (Q) := D , Q . u, f := f | u := u(f ), f (x)u(x) dx := u(f ) (f D ( )).

10.10. D ( ) D ( )

253

10.10.5. . (1) g L1,loc (RN ) . ug (f ) := f (x)g (x) dx (f D ( ))

ug . . ug g . , , : D ( ) D ( ) ug = | g . (2) . u (. . , f 0 u(f ) 0) . (3) , u m, Q RN tQ , |u(f )| t
Q

f

m,Q

(f D (Q)).

. , . (4) : (Z+ )N u : u D ( ). f D ( ) ( u)(f ) := (-1)|| u( f ). u u ( ). , . ., . ( ) . D (R) (-1) := H , H : R R R+ . () u ug , g u . . (5) u D ( ) u (f ) := u(f ) . u () u. , (. 10.9.3 (3)), .

254

. 10.

(6) E Op ( ) u D ( ). f D (E ), , u(f ). uE D (E ), u E . , D . u D ( ) E Op ( ) , E u, uE = 0. 10.10.4 (5), u , u . ( RN ) , u, u supp(u). , supp( u) supp(u). , . (7) u D ( ) f C ( ). g D ( ) f g D ( ). (f u)(g ) := u(f g ). f u f u. Tr ( ) . limTr ( ) u(f ), , u f . , u C ( ). u E ( ) := C ( ) . , u E ( ) (. 10.10.2 (3)), , u D ( ) . f C ( ) f supp(u) = 0 , || m, u m, , , u(f ) = 0. , , . (8) 1 , 2 Op (RN ) uk D ( k ). 1 в 2 , , u , fk D ( k ) u(f1 f2 ) = u1 (f1 )u2 (f2 ). u1 в u2 u1 u2 . 10.10.2 (4), , f D ( 1 в 2 ) u(f ) u1 u2 . , u(f ) = u2 (y = u1 (x
1 2

u1 (f (·, y ))) =

u2 (f (x, ·))).

: f (x, y )(u1 в u2 )(x, y ) dxdy =
1

в

2

10.10. D ( ) D ( ) =
2

255

1

f (x, y )u1 (x) dx u2 (y ) dy = f (x, y )u2 (y ) dy u1 (x) dx.
2

=
1

, supp(u1 в u2 ) = supp(u1 ) в supp(u2 ). (9) u, v D (RN ). f D (RN ) f := f +. , f C (RN в RN ). , u v Ё , , u в v f C (RN в RN ) f D (RN ). (. 10.10.10), f (u в v )( f ) (f D (RN )) . Ё u v u v . , Ё (. 9.6.17) RN (. 10.9.4 (7)) Ё . . , E (RN ) Ё (, ) . u = u, (u v ) = u v = u v . , (= ): co (supp(u v )) = co (supp(u)) + co (supp(v )). , , , Ё ((1 ) (-1) = 0 1 ( (-1) ) = 1, 1 := 1R ). u f (uf )(x) = u(x (f )), f := f f , . . f (x) := f (-x) (x RN ). u : f uf D (RN ) C (RN ), : (u)x = x u x RN . ,
+ + + +

256

. 10.

, . . T L (D (RN ), C (RN )) , , , u , T = u N u(f ) := (T )(f ) f D (R ) (. ). 10.10.6. . D ( ) D ( ) ( D ( ) D ( )# ). D ( ) (D ( ), D ( )), D ( ) D := D ( ) := (D ( ), D ( )). 10.10.7. Op (RN ). (1) D , D (Q) D ( ) Q (. . D ); (2) A D ( ) , Q A D (Q) D (Q); (3) (fn ) f (D ( ), D ) , Q , supp(fn ) Q, supp(f ) Q ( fn ) Q f (: fn f ); (4) T L (D ( ), Y ), Y , , T fn 0, fn 0; (5) (bn ) (Ё ) D (RN ), D (RN ), . . f D (RN ) u D (RN ) : bn f f ( D (RN )) bn u u N ( D (R )). (1) 10.9.6, (2) 10.9.7 = nN Qn , Qn Qn+1 n N. (3) , , 10.10.7 (2) (. 10.9.8).

10.10. D ( ) D ( )

257

(4) 10.10.7 (1) T T D (Q) Q . 10.10.2 (2) D (Q) . 10.10.7 (3). (5) , supp(bn f ) supp(f ). , g C (RN ) , bn g g RN . f (3), : bn f f. 10.10.5 (9) f D (RN ) u(f ) = (u f )(0) = lim(u (bn f ))(0) =
n

= lim((u bn ) f )(0) = lim(bn u)(f ).
n n

10.10.8. . 10.10.7 (3) Op (RN ) (m) m Z+ D (m) ( ) := C0 ( ), f , f || m . D (m) (Q) := {f D (m) ( ) : supp(f ) Q} Q · m,Q , . D (m) ( ) . , D (0) ( ) = K ( ) D ( ) = mN D (m) ( ). D (m) ( ) (fn ) m Q , supp(fn ) Q n. , D (m) ( ) m. D F ( ) := D (m) ( )
mN

, . 10.10.9. Op (RN ). (1) D ( ) , . . (= ) ; (2) D ( ) , . . D ( ) ;

258

. 10.

(3) D ( ), , , . . D ( ) ; (4) . (1) V D ( ) , VQ := V D (Q) D (Q) Q . , VQ D (Q) (. 7.1.8). (2) D (Q) Q 10.10.7 (2). 10.10.2 (2), D (Q) . 4.6.10 4.6.11, . (3) D (Q) Q . (4) g | D ( ) , D ( ) D ( ). , f D ( ) uf (g ) = 0, . . g (x)f (x) dx = 0. , g = 0. 10.5.9. 10.10.10. . (uk )kN f D ( )

u(f ) :=
k=1

uk (f ).

u

,

u =
k=1

u

k

. u 10.10.9 (1). , f D ( ) (. 10.10.5 (4)) u(f ) =

= u (-1)|| f =
k=1

u

k

(-1)|| f =

=
k=1

uk (f ). .

10.10.11. . D (. 10.9.10 10.9.11).

10.10. D ( ) D ( )

259

10.10.12. . , . . , 10.10.11, RN . , E u D( ) (uE )E E , E ( ) (k )kN . , u = k=1 k uk , uk := uEk supp(k uk ) Ek (k N). 10.10.13. . u m : u= µ ,
||m

µ M ( ). u supp(u) Q supp(u). (. 10.10.5 (7) 10.10.8) |u(f )| t
||m

f

(f D (Q))

t 0. 3.5.7 3.5.3, 10.9.4 (2) u=t
||m

= t

(-1)||
||m

( )||m , | |( · ). , (k )kN , k D ( ), Qk supp(k ) (. 10.10.12). (k u)kN k u = µk, ,
||m

µk

,

, supp(µk, ) Qk .

260

. 10.

10.10.10, ,

µ (f ) :=
k=1

µk, (f ) .

f K ( ) µ 10.10.10, :

u=
k=1

k u =
k=1 ||m

µk

,

=
||m

k=1

µk

,

=
||m

µ .

. 10.10.14. . 10.10.13 . . , , ( ) , . 10.11. 10.11.1. , L1 (RN ) := L1 (RN , C). : (1) (L1 (RN ), ), . . = 0, L1 (RN ) (f g ) = (f )(g ) (f , g L1 (RN )) (: X(L1 (RN )), . 11.6.4); (2) , , t R , f L1 (RN ) (f ) = f (t) := (f et )(0) :=
R
N

N

f (x)ei

(x,t)

dx.

(1) (2): (f )(g ) = 0. x RN , (x f g ) = (x f )(g ) = (x g )(f ).

10.11.

261

(x) := (f )-1 (x f ). : RN C. x, y RN (x + y ) = = (f g )-1 (x+y (f g )) = = (f )-1 (g )-1 · (x f y g ) = = (f )-1 (x f )(g )-1 (y g ) = = (x) (y ), . . () : X(RN ). , = et (, ) t RN . (f )(g ) = (f g ) =
R
N

(x g )f (x) dx = f (x)(g ) (x) dx =

=
R
N

(x g )f (x) dx =
R
N

= (g )
R
N

f (x) (x) dx.

, (f ) =
R
N

f (x) (x) dx (f L1 (RN )).

(2) (1): f , g f g , t RN : f g (t) = uf g (et ) =

=
R
N

f (x)g (y )et (x + y ) dxdy =
R
N

f (x)et (x) dx
R
N

g (y )et (y ) dy =
N

R

= uf (et )ug (et ) = f (t)g (t).

262

. 10.

10.11.2. . G. X(L1 (G)) () G, . . : G C, | (x)| = 1, (x + y ) = (x) (y ) (x, y G).

G := X(G) . X(L1 (G)) ((L1 (G)) , L1 (G)), G . G G . q G q : q G q (q ) C G. G G G G (= ).

10.11.3. . f L1 (RN ) f : RN C, f (t) := f (t) := (f et )(0), f . 10.11.4. . , . -, N F : L1 (RN ) C R , F f := f , (. 10.11.13). -, F F f := f , (= ) RN . : (x) := (x) := -x, (x) := 2 (x) := 2 x (x) := -2 (x) := -2 x (x RN ). , : F f (t) =
R
N

f (x)e-i

(x,t)

dx,

10.11. F2 f (t) =
R
N

263

f (x)e2

i(x,t)

dx,

F

-2

f (t) =
R
N

f (x)e-2

i(x,t)

dx.

, , , f , , F f , F f , F±2 f . F2 ( F-2 ) F-2 (, F2 ) (. 10.11.12). 10.11.5. . (1) f (x) = 1 -1 x 1 f (x) = 0 x R. f (t) = 2t-1 sin t. , k t0 > 0

|f (t)| dt
[t0 ,+) [k ,+)

|f (t)| dt =
n=k [n ,(n+1) ]

|f (t)| dt 1 = +. (n + 1)

n=k [n ,(n+1) ]

2| sin t| dt = 4 (n + 1)

n=k

, f L1 (R). / (2) f L1 (RN ) f , f f 1 . , |f (t)|
R
N

|f (x)| dx = f

1

(t RN ).

(3) f L1 (RN ) |t| + |f (t)| 0 (= ). . 5.5.9 (6) , F B (L1 (RN ), l (RN )).

264

. 10.

(4) f L1 (RN ), > 0 f (x) := f (x) (x RN ). f (t) = -N f (t / ) (t RN ).

f (t) =
R
N

f (x)et (x) dx =
-N

-N R
N

f (x)et/ (x) dx =

=

f

t

(5) F (f ) = (F f ) , (x f ) = ex f , (ex f ) = x f . (f L1 (R ), x RN .) . a b = (ab ) a, b C, , , t RN
N

F (f )(t) =
R
N

f (x)ei

(x,t)

dx =
R
N

f (x) ei

(x,t)

dx

=

=
R
N

f (x)e-i

(x,t)

dx

= (F f ) (t).

(6) f , g L1 (RN ) (f g ) = f g ;
R
N

fg =
R
N

f g.

10.11.1. : fg =
R
N

f (x)et (x) dxg (t) dt =
R
N

R

N

=
R
N

g (t)et (x) dt f (x) dx =
R
N

f g.
R
N

(7) f , f , g L1 (RN ), (f g ) = f g .

10.11. x R (f g ) (x) =
R
N

265

N

g (t)f (y )et (y )et (x) dy dt =
R
N

g (t)f (t)et (x) dt =
R
N

=
R
N

f (y )g (t)et (x + y ) dtdy =
R
N

=
R
N

f (y )g (x + y ) dy =
R
N

f (y - x)g (y ) dy = f g (x).

(8) f D (RN ) (Z+ )N F ( f ) = i|| t F f , F2 ( f ) = (2 i)|| t F2 f , (F f ) = i|| F (x f ); (F2 f ) = (2 i)|| F2 (x f )

( x := t := (·) : y RN y1 1 · . . . · yNN ). (. 10.11.4) . et = i|| t et , F ( f )(t) = et f (0) = = et f (0) = i|| t (et f )(0) = i|| t f (t). , , (F f )(t) = t1 t f (x)ei
1 R
N

(x,t)

dx =

=
R
N

f (x)ix1 ei

(x,t)

dx = F (ix1 f )(t).

fN

(9) fN (x) := exp -1/2 |x|2 x RN , = (2 )N/2 fN .

266 ,
N

. 10.

fN (t) =
k=1 R

eitk xk e-

1 2

|x|

2 k

dxk

(t RN ).

, N = 1. y R f1 (y ) =
R

e-

1 2

x2 ixy

e

dx =
R

e-
1 2

1 2

(x-iy )2 - 1 (y 2 ) 2

dx =

= f1 (y )
R

e-

(x-iy )

2

dx.

A CR R2 ( ) 1 2 . f (z ) := exp -z 2 /2 (z C) 1 2 , : 1 f (z ) dz = 2 f (z ) dz . : A=
R

e-

1 2

(x-iy )

2

dx =
R

e-

1 2

(x2 )

dx =

2 .

10.11.6. . ( , . 10.11.17 (2)) S (RN ) := := f C (RN ) : (, (Z+ )N ) |x| + x f (x) 0 ( R {p, : , (Z+ )N }, p, (f ) := x f
N

C) .

10.11.7. : (1) S (RN ) ; (2) S (RN );

10.11.

267

(3) S (RN ) ( ) {pn : n N}, pn (f ) :=
||n

(1 + | · |2 )n f

(f S (RN ))

( , |x| x RN ); (4) D (RN ) S (RN ); , D (RN ) S (RN ) S (RN ) D (RN ); (5) S (RN ) L1 (RN ). (4), . f S (RN ) D (RN ) , B N { = 1}. x R > 0 (x) := ( x),
N

f = f .

, f D (R ). > 0 , (Z+ )N . , 0 < 1 sup{ ( - 1) : , (Z+ )N } < +. , x f (x) 0 |x| +, r > 1 , |x (( (x) - 1)f (x))| < , |x| > r. , f (x) - f (x) = ( ( x) - 1)f (x) = 0 |x| -1 . , r-1 p
,

(f - f ) = sup |x (( (x) - 1)f (x))|
|x|>
-1

sup |x (( (x) - 1)f (x))| < .
|x|>r

, p, (f - f ) 0 0, . . f f S (RN ). . 10.11.8. S (RN ). f D (RN ) 10.11.5 (8), 10.11.5 (2) Ё 5.5.9 (4) t f , t f

= ( f )

f

1

K f

.

= t (x f )

K (x f )

. R
N

, f S (RN ) F D (Q) Q . 10.10.7 (4) 10.11.7 (4).

268

. 10.

10.11.9. . , S (RN ), . f S (RN ) g (x) := fN (x) = exp -1/2 |x|2 . 10.11.8 10.11.7 , f , f , g L1 (RN ) , , 10.11.5 (7), (f g ) = f g . g (x) := g (x) x RN > 0. x - 10.11.5 (4) g (t)f (t)et (x) dt =
R
N

=

1 N
R
N

f (y - x)g

y

dy =
R
N

f (y - x)g (y ) dy .

10.11.5 (9) 0, : g (0)
R
N N 2

f (t)et (x) dt = f (-x)
R
N

g (y ) dy =

= (2 ) f (x)
R
N

e-

1 2

|x|

2

dx = (2 )N f (-x).

F 2 f = (2 )N f .
2 (F2 )-1 = F 10.11.10. . F2 N N f S (R ) t R -2

.

f (-t) = (2 )N
R
N

ei

(x,t)

f (x) dx =
R
N

e2

i(x,t)

f (2 x) dx =

= (F2 (F2 f ))(t). , F2 f = F
-2

f , . , g) g ))

10.11.11. . S (RN ) Ё (= Ё). f , g S (RN ) f g S (RN ) , f g S (RN ). 10.11.5 (6) , F2 (f S (RN ) , , 10.11.10, f g = F-2 (F2 (f S (RN ).

10.11.

269

10.11.12. . F := F2 S (RN ). Ё . F-1 F-2 Ё. , : f g =
R
N

fg
R
N

(f , g S (RN )).

10.11.10 10.11.5 (5) . , 10.11.5 (7) 10.11.7 (4), (f g )(0) = (f g )(0) f g . 10.11.5 (5), : f g = (F(F
R
N

-1

f )g ) (0) = ((F

-1

f ) Fg )(0) = Ff (Fg ) .
R
N

=
R
N

Ff (Fg ) dx =
R
N

Ff F (g ) dx =

10.11.13. . 10.11.9 , F : Ff (t) =
-1

1 (2 )
N 2

f (x)ei
R
N

(x,t)

dx;

F

f (x) =

1 (2 )
N 2

f (t)e-i
R
N

(x,t)

dt.

10.11.12 -1 Ё f g := (2 )-N/2 f g (f , g L1 (RN )). F F 10.11.5 (8). F (Z+ )N : D := (2 i)-|| . 10.11.14. . S (RN ) L2 (RN ) , . 10.11.12, 4.5.10 S (RN ) L2 (RN ).

270

. 10.

10.11.15. . , 10.11.14, . ( ) L2 - Ff F-1 f f L2 (RN ) L2 (RN ). 10.11.16. . u S (RN ) := S (RN ) . u (: , . .). S (RN ), , (S (RN ), S (RN )) ( S (RN )). 10.11.17. . (1) Lp (RN ) S (RN ) 1 p +. f Lp (RN ), S (RN ), p < + 1/q + 1/p = 1. Ё 5.5.9 (4) K , K , K > 0 :
1/p q

p 1/p

B

| |p

+
RN \ B

(1 + |x|2 )N (1 + |x|2 )-N (x)

dx
1/p

K

+ (1 + | · |2 )N

RN \ B

dx (1 + |x|2 )N

p

K p1 ( ). Ё , |uf ( )| = | | f | =
R
N

f q f

p

q

K p1 ( ).

p = + .

10.11. (2) (3) . . ,

271

S (RN ) S (RN ). 10.11.7 (4), 10.11.17 (1), 10.11.7 (5) 10.10.9 (4). µ M (RN ) , n N d|µ|(x) < +. (1 + |x|2 )n
R
N

, , . (4) u S (RN ), f S (RN ) (Z+ )N , f u N S (R ) u S (RN ) 10.11.7 (2). , D u(f ) := (-1)|| uD f f S (RN ), , D u S (RN ) D u = (2 i)-|| u. (5) . u D (RN ) 10.10.5 (7) E (RN ). S (RN ) C (RN ), : u S (RN ). (6) u S (RN ). f S (RN ), u f , u f S (RN ). , u v E (RN ), u v S (RN ). (7) u D (RN ), x RN x u := (-x ) u = u -x u. u ( x), x u = u. . Ё . (8) un S (RN ) (u N) f S (RN ) u(f ) := n=1 un (f ), u S (RN ) u = n=1 un (. 10.10.10). 10.11.18. . . u S (RN ). 10.11.7 (3) 5.3.7 n N K > 0 |u(f )| K
||n

µ

(1 + | · |2 )n f

(f S (RN )).

272

. 10.

3.5.3 3.5.7, µ M (RN ) u(f ) =
||n

µ (1 + | · |2 )n f

(f S (RN )). ,

:= (-1)|| (1 + | · |2 )n µ . u=
||n

.

10.11.19. . (, , ) u S (RN ) Fu, f | Fu = Ff | u (f S (RN )).

10.11.20. . F S (RN ) S (RN ). F-1 S (RN ). . 10.11.7 (5), 10.11.12, 10.11.17 (2) 4.5.10.
10.1. , . 10.2. , , . 10.3. . 10.4. , . 10.5. . 10.6. , , (= ).

273

10.7. T , . , T . 10.8. X, Y T L (X, Y ) . , T , T (. . (X, (X, X )) (Y , (Y , Y ))). 10.9. · 1 · 2 , X , (X, · 1 ) (X, · 2 ) X . , . 10.10. S Y X . S X Y ? 10.11. (X, X # )? 10.12. (X ) . , , X := X . , (X , X ) =

(X , X );

(X , X ) =

(X , X ).

10.13. X Y , T B (X, Y ) im T = Y . , X Y. 10.14. , (X ) ( X ) . 10.15. , c0 . 10.16. p Y , T L (X , Y ) . , ext(T ( p)) T (ext( p)). 10.17. p X X X . , f ext(X p) , X = cl X + {p - f 1} - {p - f 1}. 10.18. , . 10.19. , . ?

11

11.1. 11.1.1. . e A , e = 0 ea = ae = a a A. 11.1.2. . , , F. , , , F := C. , . , A1 A2 (= ) A1 A2 , A1 A2 . A . , Ae := A в C , (a, )(b, µ) := (ab + µa + b, µ), a, b A , µ C. (a, ) Ae := a A + ||. 11.1.3. . ar A a, aar = e. al A a, al a = e. 11.1.4. , . ar = (al a)ar = al (aar ) = al e = al

11.1.

275

11.1.5. . a A a Inv(A), a . a-1 := ar = al . a-1 a. ( ) B A ( , ) A, Inv(B ) = Inv(A) B . 11.1.6. . A . a A La : x ax (x A). LA := L : a La (a A)

. L(A) B (A) L : A L(A) . x, a, b A L(ab) : x Lab (x) = abx = a(bx) = La (Lb x) = (La)(Lb)x, . . L ( L ). La = 0, 0 = La(e) = ae = a, L . L(A) Ar , A ab := ba (a, b A). R := LAr , . . Ra := Ra : x xa a A. , L(A) R(A) Z (im R) := {T B (A) : T Ra = Ra T (a A)}. , T L(A), . . T = La a A, b A La Rb (x) = axb = Rb (La (x)) = Rb La (x) T Z (R(A)). , , T Z (R(A)), a := T e La x = ax = (T e)x = Rx (T e) = (Rx T )e = (T Rx )e = = T (Rx e) = T x x A. , T = La L(A). , L(A) B (A).

276

. 11.

T = La T -1 B (A). b := T -1 e ab = La b = T b = T T -1 e = e. , ab = e aba = a T (ba) = La ba = aba = a = La e = T e. ba = e, T . , L(A) A. 5.6.3 L = sup{ La : a 1} = sup{ ab : a 1, b 1} 1.

7.4.5, , L (. . L-1 L(A) A). 11.1.7. . LA , 11.1.6, () A. 11.1.8. . , . A LA A B (A) , , A B (A). LA A L(A). . , LA , , e C, e A (. 5.6.5). , C Ce A e. 11.2. 11.2.1. . A a A. C a (: 1 res(a)), R(a, ) := -a := ( - a)-1 . Sp(a) := C \ res(a) a, Sp(a) a. , SpA (a).

11.2. 11.2.2. a A : SpA (a) = Sp LR(a, ) = R(La , )
L(A)

277

(La ) = Sp(La );

( res(a) = res(La )).

11.2.3. . ( ) (. . , , C). : e, e A C. , C A. a A. 11.2.2 8.1.11, Sp(a) = . , C , ( - a) , . . a = e. , . () = e = || e = ||, . 11.2.4. . A B A ( ). b B : SpB (b) SpA (b), SpA (b) SpB (b). b := - b Inv(B ), b Inv(A). resB (b) resA (b), . . SpB (b) = C \ res(b) C \ res(b) = SpA (b).
B A

SpB (b), b Inv(B ). (bn ), bn Inv(B ), b. t := supnN b-1 , n b =
-1 -1 = n - bm -1 bn (bm - bn

b )b

-1 n -1 m

(1 - bn b t
2

-1 m

)=
m

bn - b

.

, t < +, B a := lim b-1 . n , , ab = ba = 1, . . b Inv(B ).

278

. 11.

Inv(B ) 11.1.6, b Inv(B ). , (, , -1 -1 bn . ), b-1 +. an := b-1 n n ban = (b - bn )an + bn an b-b
n

an + b

-1 -1 n

bn b

-1 n

0.

, b . , -1 a := b 1 = an = aban a ban 0.

, - b Inv(A), . . SpA (b). ґ SpB (b), SpA (b). 11.2.5. . SpB (b) , SpB (b) = SpA (b). SpB (b) = SpB (b) SpB (b) SpA (b) SpA (b) SpB (b) 11.2.6. . : . 11.3. 11.3.1. . a A h H (Sp(a)) a. 1 h(z ) Ra h := dz . 2 i z-a Ra h A h. , , f H (Sp(a)) , a, f (a) := Ra f .

11.4.

279

11.3.2. . Ra a A n A. f (z ) := n=0 cn z ( Sp(a)), n f (a) := n=0 cn a . 11.2.3 8.2.1, 11.2.2, (LRa h)(b) = LRa h b = (Ra h)b = = 1 2 i h(z )R(a, z ) bdz = = 1 2 i 1 2 i 1 2 i h(z )R(a, z ) dz b =

h(z )R(La , z ) bdz =

h(z )R(La , z ) dz b = RLa h(b)

b A. , , RLa (H (Sp(a))) im L. , H (Sp(a)) @ @R RLa @a @ ? R L@ B (A) A H (Sp(a)) @ @R RLa @a @ ? R @ L-1 L(A) A 11.1.6 8.2.3.

11.3.3. .

280

. 11.

, 8.2 B (X ), X . 11.4. 11.4.1. . A . J A A J A, AJ J . 11.4.2. J (A) A, , . E J (A) sup
J ( A)

E = sup

Lat(A)

E,

inf

J ( A)

E = inf

Lat(A)

E,

. . J (A) Lat(A) . , 0 , A . , . 2.1.5 2.1.6. 11.4.3. J0 A. , , : A A/J0 A - A := A/J0 . J A (J ) A; J A -1 (J ) A. ab := (-1 (a)-1 (b)) a, b A, : (ab) = (a)(b) a, b A. , (J ) A(J ) = (A)(J ) (AJ ) (J ); (J ) A-1 (J ) -1 ((A)J ) = -1 (AJ ) -1 (J ).

-1

11.4.4. J A J = 0. : (1) A = J ; (2) 1 J ; / (3) J .

11.5. C (Q, C)

281

11.4.5. . J A , J A. , , . 11.4.6. , . 11.4.7. J A. (J ) (A/J ). : J A/J . , 11.4.3, -1 (J ) A. , , J -1 (J ), J = -1 (J ) 0 = (J ) = (-1 (J )) = J , A = -1 (J ) J = (-1 (J )) = (A) = A/J 1.1.6. , A/J . 11.4.6. : J0 A J0 J . , 11.4.3, (J0 ) A/J . 11.4.6 (J0 ) = 0, (J0 ) = A/J . J0 -1 (J0 ) -1 (0) = J J = J0 . (J0 ) = (A), . . A = J0 + J J0 + J0 = J0 A. , J . 11.4.8. . . J0 A. , , E J A , J0 J . E0 E 11.4.2 : sup E = {J : J E0 }. 11.4.4 sup E0 . , E 1.2.20. 11.5. C (Q, C) 11.5.1. . J C (Q, C) Q. , , Q0 := {f -1 (0) : f J }; J0 := {f C (Q, C) : int f -1 (0) Q0 }. J
0

C (Q, C), J0 J .

282

. 11.

Q1 := cl(Q \ f -1 (0)) f J0 . , , Q1 Q0 = . f J (, , ) u J , u(q ) = 1 q Q1 . , uf = f . u , q Q1 fq J , fq (q ) = 0. gq := fq fq , , , fq : x fq (x) fq , gq 0 , , gq (q ) > 0. , gq J q Q1 . (Uq )qQ1 , Uq := {x Q1 : gq (x) > 0}, Q1 . Q1 , {q1 , . . . , qn } Q1 , Q1 Uq1 . . . Uqn . g := gq1 + . . . + gqn . , g J , g (q ) > 0 q Q1 . h0 (q ) := g (q )-1 q Q1 . 10.8.20 h C (Q, R), h Q = h0 . , , u := hg . u 1 . , , J0 J . , J0 C (Q, C) . 11.5.2. J C (Q, C) , , Q0 , J = J (Q0 ) := {f C (Q, C) : q Q0 f (q ) = 0}. 9.3.14. Q0 , 11.5.1. J J (Q0 ). f J (Q0 ) n N Un := |f | 1 2n , Vn := |f | 1 n .

9.3.14, hn C (Q, R) , 0 hn 1 hn Un = 0, hn Vn = 1. fn := f hn . - int fn 1 (0) int Un Q0 , 11.5.1 fn J . , fn f .

11.6.

283

11.5.3. . C (Q, C) J (q ) := J ({q }) = {f C (Q, C) : f (q ) = 0}, q Q. 11.5.2, .

11.6. 11.6.1. A , J A , A. - A/J , -, . : A A/J , (1) A/J , = 1. a, b A , 5.1.10 (5), (a)(b) A/J = inf { a b A : (a ) = (a), (b ) = (b)} inf { a A b A : (a ) = (a), (b ) = (b)} = = (a) A/J (b) A/J . , A/J . , (1) 1. , (1)
A/J

= inf { a

A

: (a) = (1)} 1

A

= 1,

. . (1) = 1. , , = 1. . 11.6.2. . 11.6.1 A , J A, . . J A, AJ A J . 11.6.3. : A C A. = (1) = 1 ( , A C). = 0, a A 0 = (a) = (a1) = (a)(1). , (1) = 1. a A C , || > a , - a Inv(A) (. 5.6.15). 1 = (1)( - a)(( - a)-1 ). ( - a) = 0, . . (a) = . , |(a)| a 1. , = 1 |(1)| = 1, : = 1.

284

. 11.

11.6.4. . A A. A X(A), ( X(A) (A , A)) A. 11.6.5. . X(A) . 11.6.3, X(A) (A , A)- BA . (A , A)- 10.6.7. 9.4.9. 11.6.6. . A . ker , X(A) M (A) A, . X(A) A. , ker A. 2.3.11 , : A/ ker C . 11.6.1, (1) = (1) = 1, . . A/ ker C. , A/ ker . 11.4.7, , ker , . . ker M (A). m M (A) - A. , m cl m, cl m A 1 cl m ( 1 Inv(A), / 5.6.12 11.1.6). , m . - A/m : A A/m. 11.4.7 11.6.1 - A/m . 11.2.3 : A/m C. := . , X(A) ker = -1 (0) = -1 ( -1 (0)) = -1 (0) = m. ker . , ker 1 = ker 2 1 , 2 X(A). 2.3.12 C 1 = 2 . , 11.6.3, 1 = 1 (1) = 2 (1) = . 1 = 2 . 11.6.7. . 11.6.6 M (A) , M (A) X(A) ker ,

11.6.

285

A. , , 11.6.6. 11.6.8. . A X(A) . a A X(A) a() := (a). a : a(), X(A), a. a a, a A, A GA ( ). 11.6.9. . GA : a a A C (X(A), C). Sp(a) = Sp(a) = a(X(A)), a = r(a), r(a) a A.

, a A a C (X(A), C), 1 = 1 a, b A ab = ab, 11.6.3. GA . , GA . Sp(a). - a , J-a := A( - a) 11.4.4. 11.4.8 m A, m J-a . 11.6.6 m = ker . , ( - a) = 0, . . = (1) = () = (a) = a(). , Sp(a). , , Sp(a), ( - a) C (X(A), C), . . X(A), = a(). , ( - a) = 0. , - a Inv(A) : 1 = (1) = (( - a)-1 ( - a)) = (( - a)-1 )( - a) = 0. , Sp(a). Sp(a) = Sp(a).

286 Ё :

. 11. (. 11.3.3 8.1.12),

r(a) = sup{|| : Sp(a)} = sup{|| : Sp(a)} = = sup{|| : a(X(A))} = sup{|a()| : X(A)} = a , . 11.6.10. A , a2 = a 2 a A. : , t t2 , R+ , , R+ , 10.6.9 a2 = a2 = sup |a2 ()| = sup |(a2 )| =
X(A) X(A)

C (X(A),C)

= sup |(a)(a)| = sup |(a)|2 =
X(A) X(A) 2

=

sup |(a)|
X(A)

=a

2

= a 2.

: 5.6.8, r(a) = lim an 1/n . , n n , a2 = a 2 , . . r(a) = a . 10.6.9, , r(a) = a . 11.6.11. . , . GA , . . A. , GA A C (X(A), C) : A (. . A ) . 11.6.12. . a A :

11.6.

287

H (Sp(a)) = H (Sp(a)) @ @R a Ra @^ @ ? R @ GA C (X(A), C) A f (a) = f a = f (a) f H (Sp(a)). X(A). z res(a) 1 (z - a) z-a =1 1 z-a = 1 1 = . (z - a) z - (a)

, R(a, z )() = 1 1 1 () = = () = R(a, z )(). z-a z - a() z-a

, (. 5.5.9 (6)), f H (Sp(a)) f (a) = GA Ra f = GA = 1 2 i 1 2 i f (z )R(a, z ) dz 1 2 i =

f (z )GA (R(a, z )) dz = = 1 2 i

f (z )R(a, z )) dz =

f (z )R(a, z ) dz = R^(f ) = f (a). a

, , , X(A) f a() = f (a()) = f ((a)) = f (z ) 1 f (z ) dz = dz = z - (a) 2 i z-a f (z ) dz = f (a)() = f (a)(). z-a

1 = 2 i 1 = 2 i

288

. 11.

11.6.13. . A . 11.6.4 11.6.8 . X(A) e X(Ae ) : e (a, ) := (a) + (a A, C). X(Ae ) \ {e : X(A)} (a, ) := (a A, C). , X(A) (. 9.4.19), X(A) e X(Ae ) \ { } . ker = A в 0. , , . (L1 (RN ), ) 10.11.1 10.11.3 10.11.5 (3), 10.11.6 (3). 11.7. C - 11.7.1. . a A , a = a. a A , a a = aa . , a , aa = a a = 1 (. . a, a Inv(A) a-1 = a , a-1 = a). 11.7.2. A A. a A , , x, y A , a = x + iy . , x= 1 (a + a ), 2 y= 1 (a - a ). 2i

a = x - iy . . a = x1 +iy1 , (. 6.4.13) a = x + (iy1 ) = x - iy1 = x1 - iy1 . , x1 = x y1 = y . 1 1 11.7.3. . 1 = 1 1 = 1 1 = (1 1) = 1 = 1

11.7. C -

289

11.7.4. a Inv(A) a Inv(A). . aa-1 = a-1 a = 1 a Inv(A). , a-1 a = -1 aa = 1 . 11.7.3, , a Inv(A) a-1 = -1 a . a := a , . 11.7.5. Sp(a ) = Sp(a) . 11.7.6. C - . 6.4.13 a a2 = a a a a . , a a . , a = a , : a = a . a = a-1 , . . a , a 2 = a a = a-1 a = 1. , a = a = a-1 = 1. , Sp(a) Sp(a-1 ) . , Sp(a-1 ) = Sp(a)-1 . 11.7.7. C - . a A. 11.3.2

exp(a) =
n=0

an n!

=
n=0

(an ) = n!

n=0

(a )n = exp(a ). n!

h = h A, a := exp(ih), , a = exp(ih) = exp((ih) ) = exp(-ih ) = exp(-ih) = a-1 . , a C - A, 11.7.6 Sp(a) T. Sp(h), 8.2.5 (. 11.3.3) exp(i) Sp(a) T. , 1 = | exp(i)| = | exp(i Re - Im )| = exp(- Im ). Im = 0, . . R. 11.7.8. . A C -. B A C - A, b B b B . B , A.

290

. 11.

11.7.9. . C - C - . B C - ( ) C A b B . b Inv(B ), , b Inv(A). b Inv(A). 11.7.4 : b Inv(A). , b b Inv(A) (b b)-1 b b. 11.1.4 , b-1 = (b b)-1 b . , , (b b)-1 B . b b B , SpB (b b) R (. 11.7.7). 11.2.5, , SpA (b b) = SpB (b b). 0 SpA (b b), / b b Inv(B ). b Inv(B ). 11.7.10. . b C - A B - C - A, b B . SpB (b) = SpA (b). 11.7.11. . 11.7.10 11.7.9 C - . , C - , . . C -, C -. 11.8. 11.8.1. C (Q, C) f f , f (q ) := f (q ) q Q, C -. f f = sup{|f (q ) f (q )| : q Q} = sup{|f (q )|2 : q Q} = (sup |f (Q)|)2 = f 2 11.8.2. C (Q, C). C - ( ) C - C (Q, C), Q, C (Q, C). A . f A f A, f A Re f A , , Re A := {Re f : f A} C (Q, R). , Re A Q. 10.8.17 Re A C (Q, R). 11.7.2.

11.8.

291

11.8.3. . -, , -. , (A, ) (B , ) R : A B , R - A-B R A-B R , R - A B . - - , . 11.8.4. . C - A - A C (X(A), C). a A a2 = (a2 ) a2
1

R

/

2

= a aa a

1

/

2

= a a = a 2 .

11.6.10 GA A A C (X(A), C). , A X(A) . 11.6.9 11.7.7 h = h A h(X(A)) = Sp(h) R. a A. 11.7.2, : a = x + iy , x, y . , X(A) (x) R, (y ) R, GA (a) () = a () = a() = (a) = (x + iy ) = = ((x) + i(y )) = (x) - i(y ) = (x - iy ) = (a ) = = a () = GA (a )() ( X(A)). , GA - , , A C - C (X(A), C). 11.8.2, : A = C (X(A), C).

292

. 11.

11.8.5. R : A B - C - A C - B . Ra a a A. R(1) = 1, R(Inv(A)) Inv(B ). , a A SpB (R(a)) SpA (a). Ё , rA (a) rB (R(a)). a A, R(a) B , R(a) = R(a ) = R(a). A0 C -, a, B0 , R(a), A0 B0 C -. , 11.8.4 11.6.9 R(a) = R(a)
B
0

= GB0 (R(a)) = r
A
0

B

0

(R(a)) =

= rB (R(a)) rA (a) = r

(a) = GA0 (a) = a .

a A , a a . , R(a)
2

= R(a) R(a) = R(a a) a a = a 2 .

11.8.6. . a C - A Sp(a) . , , - Ra C (Sp(a), C) A , a = Ra (ISp(a) ). B C - A, a. , B a ( a a ). 11.7.10 Sp(a) = SpA (a) = SpB (a). a := GB (a) a X(B ) Sp(a) 11.6.9 , , . X(B ) Sp(a) , 9.4.11, , a . ,

R : f f a C (Sp(a), C) C (X(B ), C). 11.3.2 , 11.6.12, a = R^IC = IC a = IC a = IC
Sp(a) ^(X(B )) a

a=

a = ISp(a) a = R(ISp(a) ).

11.8.
- Ra := GB 1 R.

293

, Ra ,

-.

- - Ra (ISp(a) ) = GB 1 R(ISp(a) ) = GB 1 (a) = a.

Ra 11.8.5 , , 11.8.2, C - C (Sp(a), C) C - ( ), ISp(a) . 11.8.7. . Ra : C (Sp(a), C) A, 11.8.6, ( a A). f C (Sp(a), C), Ra (f ) f (a). 11.8.8. . f a C - A, . . f H (Sp(a)). f (a) A. f f Sp(a), Ra (f ) := Ra f Sp(a) A. , 11.8.7, f (a). 11.6.12 11.8.6. , . . , · Sp(a) , h H (Sp(a)) Sp(a), . . h Sp(a) z h z (. 8.1.21). , · Sp(a) : H (Sp(a)) C (Sp(a), C). a C - a :

294

. 11. · |Sp(a)H (Sp(a)) C (Sp(a), C) @ @ Ra @ Ra @ R @? A

. 11.9. - C - 11.9.1. . A ( ). s A A ( s S (A)), s = s(1) = 1. a A N (a) := {s(a) : s S (A)} a. 11.9.2. C (Q, C) R+ . a 0 s = s(1) = 1. , s(a) 0. z C > 0 , B (z ) := z + D a(Q). a - z , , |s(a - z )| . , |s(a) - z | = |s(a) - s(z )| , . . s(a) B (z ). , {B (z ) : B (z ) a(Q)} = cl co(a(Q)) R+ . , s(a) R+ . 11.9.3. . a C - : (1) Sp(a) N (a); (2) Sp(a) R+ N (a) R+ . B C - A, a. , B . 11.6.9 a := GB (a) a(X(B )) = SpB (a). 11.7.10, SpB (a) = Sp(a). , Sp(a) B , (a) = . 11.6.3, = (1) = 1. 7.5.11, s A . s A s(a) = . Sp(a) N (a) ( ,

11.9. - C -

295

N (a) R+ , Sp(a) R+ ). s A. , s B B . , a X(B ) Sp(a). , B C (Sp(a), C). 11.9.2 : s(a) = s B (a) 0 a 0. , Sp(a) R+ N (a) R+ , . 11.9.4. . a C - A , a Sp(a) R+ . A A+ . 11.9.5. A+ C A. , N (a + b) N (a) + N (b) N (a) = N (a) a, b A R+ . 11.9.3 1 A+ + 2 A+ A+ 1 , 2 R+ . , A+ . a A+ (-A+ ), Sp(a) = 0. , a , 11.8.6 : a = 0. 11.9.6. a C - A a+ , a- A+ , a = a+ - a- ; a+ a- = a- a+ = 0.

11.8.6. 11.9.7. . a C - A , a = b b b A. : a + , . . a = a Sp(a) R+ . (. 11.8.6) A b := a. b = b b b = a. : a = b b, a 11.9.6 : b b = u - v , uv = v u = 0 u 0, v 0 ( (AR , A+ )). : (bv ) bv = v b bv = v b bv = v (u - v )v = (v u - v 2 )v = -v 3 . v 0, v 3 0, . . (bv ) bv 0. 5.6.22 Sp((bv ) bv ) Sp(bv (bv ) ) . bv (bv ) 0.

296

. 11.

11.7.2, bv = a1 + ia2 a1 a2 . , a2 , a2 A+ (bv ) = a1 - ia2 . 1 2 11.9.5, : 0 (bv ) bv + bv (bv ) = 2 a2 + a2 0. 1 2 11.9.5, a1 = a2 = 0, . . bv = 0. , -v 3 = (bv ) bv = 0. 11.9.5 v = 0. , a = b b = u - v = u 0, . . a A+ . 11.9.8. C - A s , . . s(a ) = s(a) (a A).

11.9.7 11.9.3 a A s(a a) 0. a := a + 1 a := a + i, 0 s((a + 1) (a + 1)) = s(a a + a + a + 1) s(a) + s(a ) R; 0 s((a + i) (a + i)) = s(a a - ia + ia + 1) i(-s(a) + s(a )) R. , Im s(a) + Im s(a ) = 0; Re(-s(a)) + Re s(a ) = 0. s(a ) = Re s(a ) + i Im s(a ) = Re s(a) - i Im s(a) = s(a) . 11.9.9. s C - A. a, b A (a, b)s := s(b a). (· , ·)s A. 11.9.8 (a, b)s = s(b a) = s((a b) ) = s(a b) = (b, a) . s , ( · , ·)s . a A, 11.9.7, a a 0, , 11.9.3, (a, a)s = s(a a) 0. , ( · , ·)s .

11.9. - C -

297

11.9.10. C -. s C - A (Hs , ( · , ·)s ), xs Hs - Rs : A B (Hs ) , s(a) = (Rs (a)xs , xs )s a A {Rs (a)xs : a A} Hs . 11.9.9, (a, b)s := s(b a) a, b A, (A, ( · , ·)s ). ps (a) := (a, a)s , s : A A/ ker ps A A/ ker ps , A. , , s : A/ ker ps Hs (, ) A/ ker ps Hs , (A, ( · , ·)s ) (. 6.1.10 (4)). Hs ( · , ·)s . , , (s s a, s s b)s = (a, b)s = s(b a) (a, b A). a A ( ) La : b ab (b A). , , , La Rs (a), :
s A-A/ ker ps -Hs Rs (a) La La s s A-A/ ker ps -Hs

s

La Xs = s La . 2.3.8, , ker ps La . , La (ker ps ) ker ps . b ker ps , . . ps (b) = 0. 6.1.5, 0 (La b, La b)s = (ab, ab)s = s((ab) ab) = = s(b a ab) = (a ab, b)s ps (b)ps (a ab) = 0, . . La b ker ps . La 2.3.9, s . , s

298

. 11.

(. 5.1.3). La . , 5.3.8 s La (s )-1 s (A/ ker ps ) Hs . 4.5.10 , , Rs (a) B (Hs ). , Rs : a Rs (a) . 11.1.6 : Lab = La Lb a, b A. , s Lab = s La Lb = La s Lb = La Lb s . Lab Xs = s Lab , Lab = La Lb , Rs . , Rs , . , L1 s = s L1 = s IA = s = IA/
ker ps

s = 1s ,

. . Rs (1) = 1. s := s s . Hs (. 6.1.10 (4)) B (Hs ) (. 6.4.14 6.4.5) a, b, y A (Rs (a )s x, s y )s = (s La x, s y )s = = (La x, y )s = (a x, y )s = s(y a x) = s((ay ) x) = (x, ay )s = = (x, La y )s = (s x, s La y )s = (s x, Rs (a)s y )s = = (Rs (a) s x, s y )s . - im s Hs , Rs (a ) = Rs (a) a A, . . Rs -. xs := s 1. Rs (a)xs = Rs (a)s 1 = s La 1 = s a (a A). , {Rs (a)xs : a A} Hs . , (Rs (a)xs , xs )s = (s a, s 1)s = (a, 1)s = s(1 a) = s(a). 11.9.11. . 11.9.10 - ( : ).

11.9. - C -

299

11.9.12. . C - - C - . A C -. H - R A C - B (H ). H (Hs )sS (A) , C -, . . H := = h := (hs )s
sS (A)

Hs =
2 s Hs

S (A)

sS (A)

Hs :
sS (A)

h

< +

.

, h := (hs )sS (A) g := (gs )sS (A) H (. 6.1.10 (5) 6.1.9): (h, g ) =
sS (A)

(hs , gs )s .

, , Rs - A Hs , s S (A). 11.8.5 a A Rs (a) B (Hs ) a , h H Rs (a)h
sS (A) 2 s Hs

sS (A)

Rs (a)

2 B (Hs )

h

2 s Hs

a

2 sS (A)

h

2 s Hs

.

, R(a)h : s Rs (a)hs R(a)h H . R(a) : h R(a)h B (H ). , R : a R(a) (a A) - A. , R Rs s S (A) , R - A B (H ). , , R . a A h, g H . (R(a )h, g ) =
sS (A)

(Rs (a )hs , gs )s =

300 =
sS (A)

. 11. (Rs (a) hs , gs )s =
sS (A)

(hs , Rs (a)gs )s =

= (h, R(a)g ) = (R(a) h, g ). - h, g H R(a ) = R(a) , . - R, . . R(a) = a a A. a . 9.4.5 : a Sp(a). 11.9.3 (1) s S (A), s(a) = a . xs , - Rs (. 11.9.10), 6.1.5, a = s(a) = (Rs (a)xs , xs )s Rs (a)xs Rs (a) = Rs (a)
B (Hs ) B (Hs ) Hs

xs

Hs

xs

2 Hs

= Rs (a)

B (Hs )

(xs , xs )s =
B (Hs )

(Rs (1)xs , xs )s = Rs (a)

B (Hs )

s(1) = Rs (a)

.

R(a) Rs (a) B (Hs ) a R(a) , , 11.8.5, : a R(a) Rs (a)
B (Hs )

a.

, , a A. 11.9.7 a a . , : R(a)
2

= R(a) R(a) = R(a )R(a) = R(a a) = a a = a 2 .

.
11.1. . 11.2. A A# , (1) = 1 (Inv(A)) Inv(C). , . 11.3. Sp(a) a A U . , > 0 , Sp(a + b) U b A, b .

301

11.4. C (Q, C), C (1) ([0, 1], C) , l1 (Z) Ё

(a b)(n) :=
k=-

a

n-k bk

.

11.5. , B (X ) T , T T X . 11.6. , B (X ) T , T T X . 11.7. A . , A . 11.8. A E . E A, a A ^ a

= sup |^(E )|. a

, A A. A. 11.9. A, B , B A 1B = 1A . , B A. 11.10. A B C - ( ) T A B . , , a A f SpA (a). , SpB (T a) SpA (a) T f (a) = f (T a). 11.11. f A , A C -. , f , . . f (a a) 0 a A, , f = f (1). 11.12. C -. 11.13. , C (Q1 , C) C (Q2 , C) , Q1 Q2 . 11.14. C - . , . 11.15. . .

302

. 11.
1.

11.16. T C -, T T (a ) = (T a) a.

11.17. T H . , S H f : Sp(S ) C , T = f (S ). C -? 11.18. A, B C - - A B . , A B . 11.19. a, b C - A, ab = ba , , a b. , f (a) f (b) f R.

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319

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320

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Ar , 11.1.6 A в B , 1.1.1 Bp , 5.1.1, 5.2.11

B p , 5.1.1 BT , 5.1.3 BX , 5.1.10, 5.2.11 B (X ), 5.6.4, B (E, F ), 5.5.9 (2) B (X, Y ), 5.1.10 (7) C (Q, F ), 4.6.8 C (m) , 10.9.9 C ( ), 10.10.2 (3) D , 10.11.13 F -1 , 1.1.3 (1) F (B), 1.3.5 (1) Fp , 5.5.9 (6) F |U , 1.1.3 (5) F (U ), 1.1.3 (5) F (a1 , · ), 1.1.3 (6) F ( · , a2 ), 1.1.3 (6) F (· , ·), 1.1.3 (6) F (X, Y ), 8.3.6 G, 10.11.2 G, 10.11.2 G F , 1.1.4 H , 6.1.10 (3) H (K ), 8.1.13 H (U ), 3.1.11 IC , 8.2.10 IU , 1.1.3 (3) J (q ), 11.5.3 J (Q0 ), 11.5.2 J A, 11.4.1 K (E ), 10.9.1

K (Q), 10.9.1 K ( ), 10.9.1 LA , 11.1.6 Lp , 5.5.9 (4), 5.5.9 (6) Lp (X ), 5.5.9 LQ0 , 10.8.4 (3) L
Q0

, 10.8.4 (4)

L , 5.5.9 (5) M ( ), 10.9.4 (2) N (a), 11.9.1 Np , 5.5.9 (6) PH0 , 6.2.7 P , 8.2.10 PX1 ||X2 , 2.2.9 (4) P1 P2 , 6.2.12 R (a, ), 11.2.1 R (T , ), 5.6.13 S (A), 11.9.1 T , 7.6.2 T , 6.4.4 T , 5.1.10 (7) U , 10.5.7 U , 6.2.5 U ( ), 3.1.1 X |, 10.3.1 X , 5.1.10 (8), 10.2.11 X , 5.1.10 (8) X , 2.1.4 (2) X+ , 3.2.5 X0 , 7.6.8 X , 8.2.10 X N , 2.1.4 (4) X # , 2.2.4 XR , 3.7.1 X , 2.1.4 (4)

324
X = X1 X2 , 2.1.7 X iX , 8.4.8 X1 в X2 в . . . в XN , 2.1.4 (4) (X, ) , 10.2.11 X/X0 , 2.1.4 (6) (X/X0 , pX/X0 ), 5.1.10 (5) X Y , 10.3.3 X Y , 2.2.6 | Y , 10.3.1 B, 9.6.14 C, 2.1.2 D, 8.1.3 F, 2.1.2 N, 1.2.16 Q, 7.4.11 R, 2.1.2 R· , 3.4.1 R+ , 3.1.2 (4) R, 3.8.1 Re, 3.7.3 Re-1 , 3.7.4 T, 8.1.3 Z, 8.5.1 Z+ , 10.10.2 (2) Ae , 11.1.2 D(Q), 10.10.1 D( ), 10.10.1 D ( ), 10.10.4 D F ( ), 10.10.8 D(m) (Q), 10.10.8 D(m) ( ), 10.10.8 D(m) ( ) , 10.10.8 E( ), 10.10.2 (3) E (RN ), 10.10.5 (9) E T , 2.2.8 F, 10.11.4 Fp , 5.5.9 (6) Fr (X, Y ), 8.5.1 F(X ), 1.3.6 GA , 11.6.8 H (K ), 8.1.14 K (X ), 8.3.3 K (X, Y ), 6.6.1 L(X ), 2.2.8 L(X, Y ), 2.2.3 Lr (X, Y ), 3.2.6 (3) L , 5.5.9 (5) M( ), 10.9.3 N(µ), 10.8.11 Np (f ), 5.5.9 (4)

N , 5.5.9 (5) P(X ), 1.2.3 (4) RT , 8.2.1 Ra h, 11.3.1 S(RN ), 10.11.6 S (RN ), 10.11.16 T(X ), 9.1.2 Up , 5.2.2 UM , 5.2.4 UX , 4.1.5, 5.2.4 X , 7.6.8 0

Fu, 10.11.19 M, 5.3.9 M N, 5.3.1 M N, 5.3.1 MX , 5.1.6 M , 10.2.7 N T , 5.1.10 (3) NT , 5.1.10 (3) Ra , 11.8.7 Cl( ), 4.1.15, 9.1.4 Im f , 5.5.9 (4) Inv(A), 11.1.5 Inv(X, Y ), 5.6.12 B , 8.1.2 (4) Lat(X ), 2.1.5 LCT (X ), 10.2.3 M(A), 11.6.6 Op( ), 4.1.11, 9.1.4 Re, 2.1.2 Re f , 5.5.9 (4) Sp(a), 11.2.1 SpA (a), 11.2.1 Sp(T ), 5.6.13 T1 , 9.3.2 T2 , 9.3.5 T3 , 9.3.9 T3 1 , 9.3.15 2 T4 , 9.3.11 T(X ), 9.1.7 Tr ( ), 10.10.2 VT(X ), 10.1.5 X(A), 11.6.4 , 10.9.4 (1) (-1) , 10.10.5 (4) q , 10.9.4 (1) µ , 10.9.4 (3) µ+ , 10.8.13 µ- , 10.8.13 |µ|, 10.8.13, 10.9.4 (3)

µ , 10.9.5 µ , 10.9.4 (4) µ1 µ2 , 10.9.4 (6) µ1 в µ2 , 10.9.4 (6) µ f , 10.9.4 (7) µ , 10.9.4 (7) (U ), 10.5.1, 10.5.7 -1 (V ), 10.5.1 - F 1 (F (U )), 10.5.5 2 , 10.11.4 , 8.2.9 (T ), 5.6.13 (X, Y ), 10.3.5 (X, Y ), 10.4.4 a f , 10.9.4 (1) M , 5.2.8 , 8.2.10 abs p ol , 10.5.7 cl U , 4.1.13 co(U ), 3.1.14 co dim X , 2.2.9 (5) coim T , 2.3.1 coker T , 2.3.1 core U , 3.4.11 diam U , 4.5.3 dim X , 2.2.9 (5) dom f , 3.4.2 dom F , 1.1.2 epi f , 3.4.2 ext V , 3.6.1 fil B, 1.3.3 fr U , 4.1.13 im F , 1.1.2 inf U , 1.2.9 int U , 4.1.13 ker T , 2.3.1 lin(U ), 3.1.14 p ol , 10.5.7 rank T , 8.5.7 (2) res(a), 11.2.1 res(T ), 5.6.13 seg, 3.6.1 sup U , 1.2.9 supp(f ), 9.6.4 supp(µ), 10.8.11, 10.9.4 (5) supp(u), 10.10.5 (6) a, 11.6.8 aµ, 10.8.15 a f , 10.9.4 (1) (a, b)s , 11.9.9 c, 3.3.1 (2), 5.5.9 (3) c (E, F), 5.5.9 (3) c0 , 5.5.9 (3) c0 (E), 5.5.9 c0 (E, F), 5.5.9 (3) u, 10.10.5 (4) (p), 3.5.2 (1) | |(p), 3.7.8 U , 4.1.13 x (f ), 3.5.1 dp , 5.2.1 dx, 10.9.9 e, 10.9.4 (1), 11.1.1 f , 10.11.3 f (a), 11.3.1 {f < t}, 3.8.1 {f = t}, 3.8.1 {f t}, 3.8.1 f (T ), 8.2.1 f , 10.10.5 (9) f µ, 10.9.4 (3) f , 10.9.4 (3) f u, 10.10.5 (7) fn f , 10.10.7 (3) fn 0, 10.9.8
K

325

g , 10.11.2 g (f ), 8.2.6 h , 6.3.5 lp , lp (E), 5.5.9 (4) l , l (E), 5.5.9 (2) m, 5.5.9 (2) p q , 5.3.3 pe , 5.5.9 (5) pS , 3.8.6 p T , 5.1.3 pX/X0 , 5.1.10 (5) r (T ), 5.6.6 s, . 1.19 t , 10.11.5 (8) ug , 10.10.5 (1) u , 10.10.5 (5) u f , 10.10.5 (9) u1 u2 , 10.10.5 (8) u1 в u2 , 10.10.5 (8) x |, 10.3.1 x , 6.4.1 x , 5.1.10 (8) x , 10.11.5 (8) x+ , 3.2.12

326
x- , 3.2.12 |x|, 3.2.12 x p , 5.5.9 (4) x , 5.5.9 (2) (x), 10.11.4 X0 , 2.1.4 (6) x := x , 5.5.9 (7) eE e x x , 6.4.1 x1 x2 , x1 x2 , 1.2.12 (x1 , x2 ), 1.2.12 x | f , 5.1.11 x y , 1.2.2 x y , 5.5.6 x y , 6.2.5 | y , 10.3.1 |||y |||p , 5.5.9 (6) · , 5.1.9 · n,Q , 10.10.2 (2)

· , 5.5.9 (5) · X , 5.1.9 · |X , 5.1.9 1, 5.3.10, 10.8.4 (6) 2X , 1.2.3 (4) , 6.4.13 , 10.9.1 , 5.5.9 (6) E | · , 10.3.1 · | · , 10.3.1 · |, 10.3.1 , 1.2.2 X , 2.1.4 (5)

X , 2.1.4 (4)

h(z )R(z )dz , 8.1.20 , 11.6.8

Absolute Bipolar Theorem, 10.5.9 absolute concept, 9.4.7 absolute polar, 10.5.7 absolutely continuous measure, 10.9.4 (3) absolutely convex set, 3.1.2 (6) absolutely fundamental family of vectors, 5.5.9 (7) absorbing set, 3.4.9 addition in a vector space, 2.1.3 adherence of a filterbase, 9.4.1 adherent point, 4.1.13 adherent point of a filterbase, 9.4.1 adjoint diagram, 6.4.8 adjoint of an operator, 6.4.5 adjunction of unity, 11.1.2, affine hull, 3.1.14 affine mapping, 3.1.7, affine operator, 3.4.8 (4) affine variety, 3.1.2 (5) agreement condition, 10.9.4 (4) Akilov Criterion, 10.5.3 AlaogluBourbaki Theorem, 10.6.7 Alexandroff compactification, 9.4.22 algebra, 5.6.2 algebra of bounded operators, 5.6.5

algebra of germs of holomorphic functions, 8.1.18 algebraic basis, 2.2.9 (5) algebraic complement, 2.1.7 algebraic dual, 2.2.4 algebraic isomorphism, 2.2.5 algebraic subdifferential, 7.5.8 algebraically complementary subspace, 2.1.7 algebraically interior point, 3.4.11 algebraically isomorphic spaces, 2.2.6 algebraically reflexive space, Ex. 2.8 ambient space, 2.1.4 (3) annihilator, 7.6.8 antidiscrete topology, 9.1.8 (3) antisymmetric relation, 1.2.1 antitone mapping, 1.2.3 approximate inverse, 8.5.9 approximately invertible operator, 8.5.9 approximation property, 8.3.10 approximation property in Hilbert space, 6.6.10 arc, 4.8.2 Arens multinorm, 8.3.8 ascent, Ex. 8.10 AscoliArzel` Theorem, 4.6.10 a assignment operator

328
associate seminorm, 6.1.7 associated Hausdorff pre-Hilbert space, 6.1.10 (4) associated Hilbert space, 6.1.10 (4) associated multinormed space, 10.2.7 associated topology, 9.1.12 associativity of least upper bounds, 3.2.10 asymmetric balanced HahnBanach formula, 3.7.10 asymmetric HahnBanach formula, 3.5.5 Atkinson Theorem, 8.5.18 Automatic Continuity Principle, 7.5.5 automorphism, 10.11.4 Baire Category Theorem, 4.7.6 Baire space, 4.7.2 Balanced HahnBanach Theorem, 3.7.13 Balanced HahnBanach Theorem in a topological setting, 7.5.10 balanced set, 3.1.2 (7) balanced subdifferential, 3.7.8 Balanced Subdifferential Lemma, 3.7.9 ball, 9.6.14 Banach algebra, 5.6.3 Banach Closed Graph Theorem, 7.4.7 Banach Homomorphism Theorem, 7.4.4 Banach Inversion Stability Theorem, 5.6.12 Banach Isomorphism Theorem, 7.4.5 Banach range, 7.4.18 Banach space, 5.5.1

Banach's Fundamental Principle, 7.1.5 Banach's Fundamental Principle for a Correspondence, 7.3.7 BanachSteinhaus Theorem, 7.2.9 barrel, 10.10.9 (1) barreled normed space, 7.1.8 barreled space, 10.10.9 (1) base for a filter, 1.3.3 basic field, 2.1.2 Bessel inequality, 6.3.7 best approximation, 6.2.3 BeurlingGelfand formula, 8.1.12 (2) bilateral ideal, 8.3.3, 132; 11.6.2 bilinear form, 6.1.2 bipolar, 10.5.5 Bipolar Theorem, 10.5.8 Birkhoff Theorem, 9.2.2 Bochner integral, 5.5.9 (6) bornological space, 10.10.9 (3) boundary of an algebra, Ex. 11.8 boundary of a set, 4.1.13 boundary point, 4.1.13 bounded above, 1.2.19 bounded below, 3.2.9 bounded endomorphism algebra, 5.6.5 Bounded Index Stability Theorem, 8.5.21 bounded operator, 5.1.10 (7) bounded Radon measure, 10.9.4 (2) bounded set, 5.4.3 boundedly order complete lattice, 3.2.8 Bourbaki Criterion, 4.4.7, 46; 9.4.4 bracketing of vector spaces, 10.3.1 bra-functional, 10.3.1 bra-mapping, 10.3.1 bra-topology, 10.3.5 B -stable, 10.1.8

bump function, 9.6.19 Calkin algebra, 8.3.5 Calkin Theorem, 8.3.4 canonical embedding, 5.1.10 (8) canonical exact sequence, 2.3.5 (6) canonical operator representation, 11.1.7 canonical pro jection, 1.2.3 (4) Cantor Criterion, 4.5.6 Cantor Theorem, 4.4.9 cap, 3.6.3 (4) CauchyBunyakovskiSchwarz i inequality, 6.1.5 Cauchy filter, 4.5.2 Cauchy net, 4.5.2 CauchyWiener Integral Theorem, 8.1.7 centralizer, 11.1.6 chain, 1.2.19 character group, 10.11.2 character of a group algebra, 10.11.1 (1) character of an algebra, 11.6.4 character space of an algebra, 11.6.4 characteristic function, 5.5.9 (6) charge, 10.9.4 (3) Chebyshev metric, 4.6.8 classical Banach space, 5.5.9 (5) clopen part of a spectrum, 8.2.9 closed ball, 4.1.3 closed convex hull, 10.6.5 closed correspondence, 7.3.8 closed cylinder, 4.1.3 closed-graph correspondence, 7.3.9 closed half-space, Ex. 3.3 closed linear span, 10.5.6 closed set, 9.1.4 closed set in a metric space, 4.1.11 closure of a set, 4.1.13

329
closure operator, Ex. 1.11 coarser cover, 9.6.1 coarser filter, 1.3.6 coarser pretopology, 9.1.2 codimension, 2.2.9 (5) codomain, 1.1.2 cofinite set, Ex. 1.19 coimage of an operator, 2.3.1 coincidence of the algebraic and topological subdifferentials, 7.5.8 coinitial set, 3.3.2 cokernel of an operator, 2.3.1 comeager set, 4.7.4 commutative diagram, 2.3.3 Commutative GelfandNa ark im Theorem, 11.8.4 compact convergence, 7.2.10 Compact Index Stability Theorem, 8.5.20 compact-open topology, 8.3.8 compact operator, 6.6.1 compact set, 9.4.2 compact set in a metric space, 4.4.1 compact space, 9.4.7 compact topology, 9.4.7 compactly-supported distribution, 10.10.5 (6) compactly-supported function, 9.6.4 compactum, 9.4.17 compatible topology, 10.4.1 complementary pro jection, 2.2.9 (4) complementary subspace, 7.4.9 Complementation Principle, 7.4.10 complemented subspace, 7.4.9 complement of an orthopro jection, 6.2.10 complement of a pro jection, 2.2.9 (4) complete lattice, 1.2.13

330
complete metric space, 4.5.5 complete set, 4.5.14 completely regular space, 9.3.15 completion, 4.5.13 complex conjugate, 2.1.4 (2) complex distribution, 10.10.5 (5) complex plane, 8.1.3 complex vector space, 2.1.3 complexification, 8.4.8 complexifier, 3.7.4 composite correspondence, 1.1.4 Composite Function Theorem, 8.2.8 composition, 1.1.4 Composition Spectrum Theorem, 5.6.22 cone, 3.1.2 (4) conical hull, 3.1.14 conical segment, 3.1.2 (9) conical slice, 3.1.2 (9) conjugate distribution, 10.10.5 (5) conjugate exponent, 5.5.9 (4) conjugate-linear functional, 2.2.4 conjugate measure, 10.9.4 (3) connected elementary compactum, 4.8.5 connected set, 4.8.4 constant function, 5.3.10, 64; 10.8.4 (6) Continuous Extension Principle, 7.5.11 continuous function at a point, 4.2.2, 43; 9.2.5 Continuous Function Recovery Lemma, 9.3.12 continuous functional calculus, 11.8.7 continuous mapping of a metric space, 4.2.2 continuous mapping of a topological space, 9.2.4 continuous partition of unity, 9.6.6

contour integral, 8.1.20 conventional summation, 5.5.9 (4) convergent filterbase, 4.1.16 convergent net, 4.1.17 convergent sequence space, 3.3.1 (2) convex combination, 3.1.14 convex correspondence, 3.1.7 convex function, 3.4.4 convex hull, 3.1.14 convex set, 3.1.2 (8) convolution algebra, 10.9.4 (7) convolution of a measure and a function, 10.9.4 (7) convolution of distributions, 10.10.5 (9) convolution of functions, 9.6.17 convolution of measures, 10.9.4 (7) convolutive distributions, 10.10.5 (9) coordinate pro jection, 2.2.9 (3) coordinatewise operation, 2.1.4 (4) core, 3.4.11 correspondence, 1.1.1 correspondence in two arguments, 1.1.3 (6) correspondence onto, 1.1.3 (3) coset, 1.2.3 (4) coset mapping, 1.2.3 (4) countable convex combination, 7.1.3 Countable Partition Theorem, 9.6.20 countable sequence, 1.2.16 countably normable space, 5.4.1 cover of a set, 9.6.1 C -algebra, 6.4.13 C -subalgebra, 11.7.8 DavisFigielSzankowski Counterexample, 8.3.14

de Branges Lemma, 10.8.16 decomplexification, 6.1.10 (2) decomposition reduces an operator, 2.2.9 (4) decreasing mapping, 1.2.3 Dedekind complete vector lattice, 3.2.8 deficiency, 8.5.1 delta-function, 10.9.4 (1) delta-like sequence, 9.6.15 -like sequence, 9.6.15 -sequence, 9.6.15 dense set, 4.5.10 denseness, 4.5.10 density of a measure, 10.9.4 (3) derivative in the distribution sense, 10.10.5 (4) derivative of a distribution, 10.10.5 (4) descent, Ex. 8.10 diagonal, 1.1.3 (3) diagram prime, 7.6.5 Diagram Prime Principle, 7.6.7 diagram star, 6.4.8 Diagram Star Principle, 6.4.9 diameter, 4.5.3 Diedonn` Lemma, 9.4.18 e dimension, 2.2.9 (5) Dini Theorem, 7.2.10 Dirac measure, 10.9.4 (1) direct polar, 7.6.8, 116; 10.5.1 direct sum decomposition, 2.1.7 direct sum of vector spaces, 2.1.4 (5) directed set, 1.2.15 direction, 1.2.15 directional derivative, 3.4.12 discrete element, 3.3.6 Discrete KrenRutman Theorem, i 3.3.8 discrete topology, 9.1.8 (4) disjoint measures, 10.9.4 (3) disjoint sets, 4.1.10

331
distance, 4.1.1 distribution, 10.10.4 distribution applies to a function, 10.10.5 (7) Distribution Localization Principle, 10.10.12 distribution of finite order, 10.10.5 (3) distribution size at most m, 10.10.5 (3) distribution of slow growth, 10.11.16 distributions admitting convolution, 10.10.5 (9) distributions convolute, 10.10.5 (9) division algebra, 11.2.3 domain, 1.1.2 Dominated Extension Theorem, 3.5.4 Double Prime Lemma, 7.6.6 double prime mapping, 5.1.10 (8) double sharp, Ex. 2.7 downward-filtered set, 1.2.15 dual diagram, 7.6.5 dual group, 10.11.2 dual norm of a functional, 5.1.10 (8) dual of a locally convex space, 10.2.11 dual of an operator, 7.6.2 duality bracket, 10.3.3 duality pair, 10.3.3 dualization, 10.3.3 Dualization Theorem, 10.3.9 DunfordHille Theorem, 8.1.3 Dunford Theorem, 8.2.7 (2) DvoretzkyRogers Theorem, 5.5.9 (7) dyadic-rational point, 9.3.13 effective domain of definition, 3.4.2

332
Eidelheit Separation Theorem, 3.8.14 eigenvalue, 6.6.3 (4) eigenvector, 6.6.3 element of a set, 1.1.3 (4) elementary compactum, 4.8.5 endomorphism, 2.2.1, 12; 8.2.1 endomorphism algebra, 2.2.8, 13; 5.6.5 endomorphism space, 2.2.8 Enflo counterexample, 8.3.12 entourage, 4.1.5 envelope, Ex. 1.11 epigraph, 3.4.2 epimorphism, 2.3.1 -net, 8.3.2 -perpendicular, 8.4.1 -Perpendicular Lemma, 8.4.1 Equicontinuity Principle, 7.2.4 equicontinuous set, 4.2.8 equivalence, 1.2.2 equivalence class, 1.2.3 (4) equivalent multinorms, 5.3.1 equivalent seminorms, 5.3.3 estimate for the diameter of a spherical layer, 6.2.1 Euler identity, 8.5.17 evaluation mapping, 10.3.4 (3) everywhere-defined operator, 2.2.1 everywhere dense set, 4.7.3 (3) exact sequence, 2.3.4 exact sequence at a term, 2.3.4 exclave, 8.2.9 expanding mapping, Ex. 4.14 extended function, 3.4.2 extended real axis, 3.8.1 extended reals, 3.8.1 extension of an operator, 2.3.6 exterior of a set, 4.1.13 exterior point, 4.1.13 Extreme and Discrete Lemma, 3.6.4 extreme point, 3.6.1 extreme set, 3.6.

face, 3.6.1 factor set, 1.2.3 (4) faithful representation, 8.2.2 family, 1.1.3 (4) filter, 1.3.3 filterbase, 1.3.1 finer cover, 9.6.1 finer filter, 1.3.6 finer multinorm, 5.3.1 finer pretopology, 9.1.2 finer seminorm, 5.3.3 finest multinorm, 5.1.10 (2) finite complement filter, 5.5.9 (3) finite descent, Ex. 8.10 finite-rank operator, 6.6.8, 97; 8.3.6 finite-valued function, 5.5.9 (6) first category set, 4.7.1 first element, 1.2.6 fixed point, Ex. 1.11 flat, 3.1.2 (5) formal duality, 2.3.15 Fourier coefficient family, 6.3.15 FourierPlancherel transform, 10.11.15 FourierSchwartz transform, 10.11.19 Fourier series, 6.3.16 Fourier transform of a distribution, 10.11.19 Fourier transform of a function, 10.11.3 Fourier transform relative to a basis, 6.3.15 Frґ het space, 5.5.2 ec Fredholm Alternative, 8.5.6 Fredholm index, 8.5.1 Fredholm operator, 8.5.1 Fredholm Theorem, 8.5.8 frontier of a set, 4.1.13 from A into/to B , 1.1.1

Fubini Theorem for distributions, 10.10.5 (8) Fubini Theorem for measures, 10.9.4 (6) full subalgebra, 11.1.5 fully norming set, 8.1.1 Function Comparison Lemma, 3.8.3 function of class C (m) , 10.9.9 function of compact support, 9.6.4 Function Recovery Lemma, 3.8.2 functor, 10.9.4 (4) fundamental net, 4.5.2 fundamental sequence, 4.5.2 fundamentally summable family of vectors, 5.5.9 (7 gauge, 3.8.6 gauge function, 3.8.6 Gauge Theorem, 3.8.7 -correspondence, 3.1.6 -hull, 3.1.11 -set, 3.1.1 GelfandDunford Theorem in an operator setting, 8.2.3 GelfandDunford Theorem in an algebraic setting, 11.3.2 Gelfand formula, 5.6.8 GelfandMazur Theorem, 11.2.3 GelfandNamarkSegal i construction, 11.9.11 Gelfand Theorem, 7.2.2 Gelfand transform of an algebra, 11.6.8 Gelfand transform of an element, 11.6.8 Gelfand Transform Theorem, 11.6.9 general form of a compact operator in Hilbert space, 6.6.9 general form of a linear functional in Hilbert space, 6.4.2

333
general form of a weakly continuous functional, 10.3.10 general position, Ex. 3.10 generalized derivative in the Sobolev sense, 10.10.5 (4) Generalized Dini Theorem, 10.8.6 generalized function, 10.10.4 Generalized RieszSchauder Theorem, 8.4.10 generalized sequence, 1.2.16 Generalized Weierstrass Theorem, 10.9.9 germ, 8.1.14 GNS-construction, 11.9.11 GNS-Construction Theorem, 11.9.10 gradient mapping, 6.4.2 GramSchmidt orthogonalization process, 6.3.14 graph norm, 7.4.17 Graph Norm Principle, 7.4.17 greatest element, 1.2.6 greatest lower bound, 1.2.9 Grothendieck Criterion, 8.3.11 Grothendieck Theorem, 8.3.9 ground field, 2.1.3 ground ring, 2.1.1 group algebra, 10.9.4 (7) group character, 10.11.1 Haar integral, 10.9.4 (1) HahnBanach Theorem, 3.5.3 HahnBanach Theorem in analytical form, 3.5.4 HahnBanach Theorem in geometric form, 3.8.12 HahnBanach Theorem in subdifferential form, 3.5.4 Hamel basis, 2.2.9 (5) Hausdorff Completion Theorem, 4.5.12 Hausdorff Criterion, 4.6.7

334
Hausdorff metric, Ex. 4.8 Hausdorff multinorm, 5.1.8 Hausdorff multinormed space, 5.1.8 Hausdorff space, 9.3.5 Hausdorff Theorem, 7.6.12 Hausdorff topology, 9.3.5 H -closed space, Ex. 9.10 Heaviside function, 10.10.5 (4) HellingerToeplitz Theorem, 6.5.3 hermitian element, 11.7.1 hermitian form, 6.1.1 hermitian operator, 6.5.1 hermitian state, 11.9.8 Hilbert basis, 6.3.8 Hilbert cube, 9.2.17 (2) Hilbert dimension, 6.3.13 Hilbert identity, 5.6.19 Hilbert isomorphy, 6.3.17 HilbertSchmidt norm, Ex. 8.9 HilbertSchmidt operator, Ex. 8.9 HilbertSchmidt Theorem, 6.6.7 Hilbert space, 6.1.7 Hilbert-space isomorphism, 6.3.17 Hilbert sum, 6.1.10 (5) HЁlder inequality, 5.5.9 (4) o holey disk, 4.8.5 holomorphic function, 8.1.4 Holomorphy Theorem, 8.1.5 homeomorphism, 9.2.4 homomorphism, 7.4.1 HЁrmander transform, Ex. 3.19 o hyperplane, 3.8.9 hypersubspace, 3.8. ideal, 11.4.1 Ideal and Character Theorem, 11.6.6 ideal correspondence, 7.3.3 Ideal Correspondence Lemma, 7.3.4

Ideal Correspondence Principle, 7.3.5 Ideal HahnBanach Theorem, 7.5.9 ideally convex function, 7.5.4 ideally convex set, 7.1.3 idempotent operator, 2.2.9 (4) identical embedding, 1.1.3 (3) identity, 10.9.4 identity element, 11.1.1 identity mapping, 1.1.3 (3) identity relation, 1.1.3 (3) image, 1.1.2 image of a filterbase, 1.3.5 (1) image of a set, 1.1.3 (5) image of a topology, 9.2.12 image topology, 9.2.12 Image Topology Theorem, 9.2.11 imaginary part of a function, 5.5.9 (4) increasing mapping, 1.2.3 (5) independent measure, 10.9.4 (3) index, 8.5.1 indicator function, 3.4.8 (2) indiscrete topology, 9.1.8 (3) induced relation, 1.2.3 (1) induced topology, 9.2.17 (1) inductive limit topology, 10.9.6 inductive set, 1.2.19 infimum, 1.2.9 infinite-rank operator, 6.6.8 infinite set, 5.5.9 (3) inner product, 6.1.4 integrable function, 5.5.9 (4) integral, 5.5.9 (4) integral with respect to a measure, 10.9.3 interior of a set, 4.1.13 interior point, 4.1.13 intersection of topologies, 9.1.14 interval, 3.2.15 Interval Addition Lemma, 3.2.15

invariant subspace, 2.2.9 (4) inverse-closed subalgebra, 11.1.5 inverse image of a multinorm, 5.1.10 (3) inverse image of a preorder, 1.2.3 (3) inverse image of a seminorm, 5.1.4 inverse image of a set, 1.1.3 (5) inverse image of a topology, 9.2.9 inverse image of a uniformity, 9.5.5 (3) inverse image topology, 9.2.9 Inverse Image Topology Theorem, 9.2.8 inverse of a correspondence, 1.1.3 (1) inverse of an element in an algebra, 11.1.5 Inversion Theorem, 10.11.12 invertible element, 11.1.5 invertible operator, 5.6.10 involution, 6.4.13 involutive algebra, 6.4.13 irreducible representation, 8.2.2 irreflexive space, 5.1.10 (8) isolated part of a spectrum, 8.2.9 isolated point, 8.4.7 isometric embedding, 4.5.11 isometric isomorphism of algebras, 11.1.8 isometric mapping, 4.5.11 isometric representation, 11.1.8 isometric -isomorphism, 11.8.3 isometric -representation, 11.8.3 isometry into, 4.5.11 isometry onto, 4.5.11 isomorphism, 2.2.5 isotone mapping, 1.2 James Theorem, 10.7.5 Jensen inequality, 3.4.5 join, 1.2.12 Jordan arc, 4.8.2 Jordan Curve Theorem, 4.8.3 juxtaposition, 2.2.

335

Kakutani Criterion, 10.7.1 Kakutani Lemma, 10.8.7 Kakutani Theorem, 7.4.11 (3) Kantorovich space, 3.2.8 Kantorovich Theorem, 3.3.4 KaplanskyFukamija Lemma, 11.9.7 Kato Criterion, 7.4.19 kernel of an operator, 2.3.1 ket-mapping, 10.3.1 ket-topology, 10.3.5 Kolmogorov Normability Criterion, 5.4.5 KrenMilman Theorem, 10.6.5 i KrenMilman Theorem i in subdifferential form, 3.6.5 KrenRutman Theorem, 3.3.5 i Krull Theorem, 11.4.8 KuratowskiZorn Lemma, 1.2.20 K -space, 3.2.8 K -ultrametric, 9.5.13 last element, 1.2.6 lattice, 1.2.12 lear trap map, 3.7.4 least element, 1.2.6 Lebesgue measure, 10.9.4 (1) Lebesgue set, 3.8.1 Lefschetz Lemma, 9.6.3 left approximate inverse, 8.5.9 left Haar measure, 10.9.4 (1) left inverse of an element in an algebra, 11.1.3 lemma on continuity of a convex function, 7.5.1 lemma on the numeric range of a hermitian element, 11.9.3 level set, 3.8.1 Levy Pro jection Theorem, 6.2.2 limit of a filterbase, 4.1.16

336
Lindenstrauss space, 5.5.9 (5) LindenstraussTzafriri Theorem, 7.4.11 (3) linear change of a variable under the subdifferential sign, 3.5.4 linear combination, 2.3.12 linear correspondence, 2.2.1, 12; 3.1.7 linear functional, 2.2.4 linear operator, 2.2.1 linear representation, 8.2.2 linear set, 2.1.4 (3) linear space, 2.1.4 (3) linear span, 3.1.14 linear topological space, 10.1.3 linear topology, 10.1.3 linearly independent set, 2.2.9 (5) linearly-ordered set, 1.2.19 Lions Theorem of Supports, 10.10.5 (9) Liouville Theorem, 8.1.10 local data, 10.9.11 locally compact group, 10.9.4 (1) locally compact space, 9.4.20 locally compact topology, 9.4.20 locally convex space, 10.2.9 locally convex topology, 10.2.1 locally finite cover, 9.6.2 locally integrable function, 9.6.17 locally Lipschitz function, 7.5.6 loop, 4.8.2 lower bound, 1.2.4 lower limit, 4.3.5 lower right Dini derivative, 4.7.7 lower semicontinuous, 4.3.3 L2 -Fourier transform, 10.11.15 MackeyArens Theorem, 10.4.5 Mackey Theorem, 10.4.6 Mackey topology, 10.4.4 mapping, 1.1.3 (3) massive subspace, 3.3.2 matrix form, 2.2.9 (4)

maximal element, 1.2.10 maximal ideal, 11.4.5 maximal ideal space, 11.6.7 Maximal Ideal Theorem, 11.5.3 Mazur Theorem, 10.4.9 meager set, 4.7.1 measure, 10.9.3 Measure Localization Principle, 10.9.10 measure space, 5.5.9 (4) meet, 1.2.12 member of a set, 1.1.3 (4) metric, 4.1.1 metric space, 4.1.1 metric topology, 4.1.9 metric uniformity, 4.1.5 Metrizability Criterion, 5.4.2 metrizable multinormed space, 5.4.1 minimal element, 1.2.10 Minimal Ideal Theorem, 11.5.1 MinkowskiAscoliMazur Theorem, 3.8.12 Minkowski functional, 3.8.6 Minkowski inequality, 5.5.9 (4) minorizing set, 3.3.2 mirror, 10.2.7 module, 2.1.1 modulus of a scalar, 5.1.10 (4) modulus of a vector, 3.2.12 mollifier, 9.6.14 mollifying kernel, 9.6.14 monomorphism, 2.3.1 monoquotient, 2.3.11 Montel space, 10.10.9 (2) Moore subnet, 1.3.5 (2) morphism, 8.2.2, 126; 11.1.2 morphism representing an algebra, 8.2.2 Motzkin formula, 3.1.13 (5) multimetric, 9.5.9 multimetric space, 9.5.9 multimetric uniformity, 9.5.9

multimetrizable topological space, 9.5.10 multimetrizable uniform space, 9.5.10 multinorm, 5.1.6 Multinorm Comparison Theorem, 5.3.2 multinorm summable family of vectors, 5.5.9 (7) multinormed space, 5.1.6 multiplication formula, 10.11.5 multiplication of a germ by a complex number, 8.1.16 multiplicative linear operator, 8.2.2 natural order, 3.2.6 (1) negative part, 3.2.12 neighborhood about a point, 9.1.1 (2) neighborhood about a point in a metric space, 4.1.9 neighborhood filter, 4.1.10 neighborhood filter of a set, 9.3.7 neighborhood of a set, 8.1.13 (2), 124; 9.3.7 Nested Ball Theorem, 4.5.7 nested sequence, 4.5.7 net, 1.2.16 net having a subnet, 1.3.5 (2) net lacking a subnet, 1.3.5 (2) Neumann series, 5.6.9 Neumann Series Expansion Theorem, 5.6.9 neutral element, 2.1.4 (3), 11; 10.9.4 Nikol ski Criterion, 8.5.22 i Noether Criterion, 8.5.14 nonarchimedean element, 5.5.9 (5) nonconvex cone, 3.1.2 (4) Nonempty Subdifferential Theorem, 3.5.8

337
non-everywhere-defined operator, 2.2.1 nonmeager set, 4.7.1 nonpointed cone, 3.1.2 (4) nonreflexive space, 5.1.10 (8) norm, 5.1.9 norm convergence, 5.5.9 (7) normable multinormed space, 5.4.1 normal element, 11.7.1 normal operator, Ex. 8.17 normal space, 9.3.11 normalized element, 6.3.5 normally solvable operator, 7.6.9 normative inequality, 5.1.10 (7) normed algebra, 5.6.3 normed dual, 5.1.10 (8) normed space, 5.1.9 normed space of bounded elements, 5.5.9 (5) norming set, 8.1.1 norm-one element, 5.5.6 nowhere dense set, 4.7.1 nullity, 8.5.1 numeric family, 1.1.3 (4) numeric function, 9.6.4 numeric range, 11.9.1 numeric set, 1.1.3 ( one-point compactification, 9.4.22 one-to-one correspondence, 1.1.3 (3) open ball, 4.1.3 open ball of RN , 9.6.16 open correspondence, 7.3.12 Open Correspondence Principle, 7.3.13 open cylinder, 4.1.3 open half-space, Ex. 3.3 Open Mapping Theorem, 7.4.6 open segment, 3.6.1 open set, 9.1.4 open set in a metric space, 4.1.11

338
openness at a point, 7.3.6 operator, 2.2.1 operator ideal, 8.3.3 operator norm, 5.1.10 (7) operator representation, 8.2.2 order, 1.2.2 order by inclusion, 1.3.1 order compatible with vector structure, 3.2.1 order ideal, 10.8.11 order of a distribution, 10.10.5 (3) ordered set, 1.2.2 ordered vector space, 3.2.1 ordering, 1.2.2 ordering cone, 3.2.4 oriented envelope, 4.8.8 orthocomplement, 6.2.5 orthogonal complement, 6.2.5 orthogonal family, 6.3.1 orthogonal orthopro jections, 6.2.12 orthogonal set, 6.3.1 orthogonal vectors, 6.2.5 orthonormal family, 6.3.6 orthonormal set, 6.3.6 orthonormalized family, 6.3.6 orthopro jection, 6.2.7 Orthopro jection Summation Theorem, 6.3.3 Orthopro jection Theorem, 6.2.10 Osgood Theorem, 4.7. pair-dual space, 10.3.3 pairing, 10.3.3 pairwise orthogonality of finitely many orthopro jections, 6.2.14 paracompact space, 9.6.9 Parallelogram Law, 6.1.8 Parseval identity, 6.3.16, 89; 10.11.12 part of an operator, 2.2.9 (4) partial correspondence, 1.1.3 (6) parti parti parti parti parti

al operator, 2.2.1 al order, 1.2.2 al sum, 5.5.9 (7) tion of unity, 9.6.6 tion of unity subordinate to a cover, 9.6.7 patch, 10.9.11 perforated disk, 4.8.5 periodic distribution, 10.11.17 (7) Pettis Theorem, 10.7.4 Phillips Theorem, 7.4.13 Plancherel Theorem, 10.11.14 point finite cover, 9.6.2 point in a metric space, 4.1.1 point in a space, 2.1.4 (3) point in a vector space, 2.1.3 pointwise convergence, 9.5.5 (6) pointwise operation, 2.1.4 (4) polar, 7.6.8, 116; 10.5.1 Polar Lemma, 7.6.11 polarization identity, 6.1.3 Pontryaginvan Kampen Duality Theorem, 10.11.2 poset, 1.2.2 positive cone, 3.2.5 positive definite inner product, 6.1.4 positive distribution, 10.10.5 (2) positive element of a C -algebra, 11.9.4 positive form on a C -algebra, Ex. 11.11 positive hermitian form, 6.1.4 positive matrix, Ex. 3.13 positive operator, 3.2.6 (3) positive part, 3.2.12 positive semidefinite hermitian form, 6.1.4 positively homogeneous functional, 3.4.7 (2) powerset, 1.2.3 (4) precompact set, Ex. 9.16 pre-Hilbert space, 6.1.7

preimage of a multinorm, 5.1.10 (3) preimage of a seminorm, 5.1.4 preimage of a set, 1.1.3 (5) preintegral, 5.5.9 (4) preneighborhood, 9.1.1 (2) preorder, 1.2.2 preordered set, 1.2.2 preordered vector space, 3.2.1 presheaf, 10.9.4 (4) pretopological space, 9.1.1 (2) pretopology, 9.1.1 primary Banach space, Ex. 7.17 prime mapping, 6.4.1 Prime Theorem, 10.2.13 Principal Theorem of the Holomorphic Functional Calculus, 8.2.4 product, 4.3.2 product of a distribution and a function, 10.10.5 (7) product of germs, 8.1.16 product of sets, 1.1.1, 1; 2.1.4 (4) product of topologies, 9.2.17 (2) product of vector spaces, 2.1.4 (4) product topology, 4.3.2, 44; 9.2.17 (2) pro jection onto X1 along X2 , 2.2.9 (4) pro jection to a set, 6.2.3 proper ideal, 11.4.5 pseudometric, 9.5.7 p-sum, 5.5.9 (6) p-summable family, 5.5.9 (4) punctured compactum, 9.4.21 pure subalgebra, 11.1.5 Pythagoras Lemma, 6.2.8 Pythagoras Theorem, 6.3. quasinilp quotient quotient quotient otent, Ex. 8.18 mapping, 1.2.3 (4) multinorm, 5.3.11 of a mapping, 1.2.3 (4)

339
quotient of a seminormed space, 5.1.10 (5) quotient seminorm, 5.1.10 (5) quotient set, 1.2.3 (4) quotient space of a multinormed space, 5.3.11 quotient vector space, 2.1.4 (6 radical, 11.6.11 Radon F-measure, 10.9.3 RadonNikodym Theorem, ґ 10.9.4 (3) range of a correspondence, 1.1.2 rank, 8.5.7 (2) rare set, 4.7.1 Rayleigh Theorem, 6.5.2 real axis, 2.1.2 real carrier, 3.7.1 real C-measure, 10.9.4 (3) real distribution, 10.10.5 (5) real hyperplane, 3.8.9 real measure, 10.9.4 real part map, 3.7.2 real part of a function, 5.5.9 (4) real part of a number, 2.1.2 real subspace, 3.1.2 (3) real vector space, 2.1.3 realification, 3.7.1 realification of a pre-Hilbert space, 6.1.10 (2) realifier, 3.7.2 reducible representation, 8.2.2 refinement, 9.6.1 reflection of a function, 10.10.5 reflexive relation, 1.2.1 reflexive space, 5.1.10 (8) regular distribution, 10.10.5 (1) regular operator, 3.2.6 (3) regular space, 9.3.9 regular value of an operator, 5.6.13 relation, 1.1.3 (2) relative topology, 9.2.17 (1)

340
relatively compact set, 4.4.4 removable singularity, 8.2.5 (2) representation, 8.2.2 representation space, 8.2.2 reproducing cone, Ex. 7.12 residual set, 4.7.4 resolvent of an element of an algebra, 11.2.1 resolvent of an operator, 5.6.13 resolvent set of an operator, 5.6.13 resolvent value of an element of an algebra, 11.2.1 resolvent value of an operator, 5.6.13 restriction, 1.1.3 (5) restriction of a distribution, 10.10.5 (6) restriction of a measure, 10.9.4 (4) restriction operator, 10.9.4 (4) reversal, 1.2.5 reverse order, 1.2.3 (2) reverse polar, 7.6.8, 116; 10.5.1 reversed multiplication, 11.1.6 Riemann function, 4.7.7 RiemannLebesgue Lemma, 10.11.5 (3) Riemann Theorem on Series, 5.5.9 (7) Riesz Criterion, 8.4.2 Riesz Decomposition Property, 3.2.16 RieszDunford integral, 8.2.1 RieszDunford Integral Decomposition Theorem, 8.2.13 RieszDunford integral in an algebraic setting, 11.3.1 RieszFisher Completeness Theorem, 5.5.9 (4) RieszFisher Isomorphism Theorem, 6.3.16

Riesz idempotent, 8.2.11 RieszKantorovich Theorem, 3.2.17 Riesz operator, Ex. 8.15 Riesz Prime Theorem, 6.4.1 Riesz pro jection, 8.2.11 RieszSchauder operator, Ex. 8.11 RieszSchauder Theorem, 8.4.8 Riesz space, 3.2.7 Riesz Theorem, 5.3.5 right approximate inverse, 8.5.9 right Haar measure, 10.9.4 (1) right inverse of an element in an algebra, 11.1.3 R-measure, 10.9.4 (3) rough draft, 4.8.8 row-by-column rule, 2.2.9 (4 salient cone, 3.2.4 Sard Theorem, 7.4.12 scalar, 2.1.3 scalar field, 2.1.3 scalar multiplication, 2.1.3 scalar product, 6.1.4 scalar-valued function, 9.6.4 Schauder Theorem, 8.4.6 Schwartz space of distributions, 10.11.16 Schwartz space of functions, 10.11.6 Schwartz Theorem, 10.10.10 second dual, 5.1.10 (8) selfadjoint operator, 6.5.1 semi-extended real axis, 3.4.1 semi-Fredholm operator, Ex. 8.13 semi-inner product, 6.1.4 semimetric, 9.5.7 semimetric space, 9.5.7 seminorm, 3.7.6 seminorm associated with a positive element, 5.5.9 (5)

seminormable space, 5.4.6 seminormed space, 5.1.5 semisimple algebra, 11.6.11 separable space, 6.3.14 separated multinorm, 5.1.8 separated multinormed space, 5.1.8 separated topological space, 9.3.2 separated topology, 9.3.2 separating hyperplane, 3.8.13 Separation Theorem, 3.8.11 Sequence Prime Principle, 7.6.13 sequence space, 3.3.1 (2) Sequence Star Principle, 6.4.12 series sum, 5.5.9 (7) sesquilinear form, 6.1.2 set absorbing another set, 3.4.9 set in a space, 2.1.4 (3) set lacking a distribution, 10.10.5 (6) set lacking a functional, 10.8.13 set lacking a measure, 10.9.4 (5) set of arrival, 1.1.1 set of departure, 1.1.1 set of second category, 4.7.1 set supporting a measure, 10.9.4 (5) set that separates the points of another set, 10.8.9 set void of a distribution, 10.10.5 (6) set void of a functional, 10.8.13 set void of a measure, 10.9.4 (5) setting in duality, 10.3.3 setting primes, 7.6.5 sheaf, 10.9.11 shift, 10.9.4 (1) Shilov boundary, Ex. 11.8 Shilov Theorem, 11.2.4 short sequence, 2.3.5 -compact, 10.9.8 signed measure, 10.9.4 (3) simple convergence, 9.5.5 (6)

341
simple function, 5.5.9 (6) simple Jordan loop, 4.8.2 single-valued correspondence, 1.1.3 (3) Singularity Condensation Principle, 7.2.12 Singularity Fixation Principle, 7.2.11 skew field, 11.2.3 slowly increasing distribution, 10.11.16 smooth function, 9.6.13 smoothing process, 9.6.18 Snowflake Lemma, 2.3.16 space countable at infinity, 10.9.8 space of bounded elements, 5.5.9 (5) space of bounded functions, 5.5.9 (2) space of bounded operators, 5.1.10 (7) space of compactly-supported distributions, 10.10.5 (9) space of convergent sequences, 5.5.9 (3) space of distributions of order at most m, 10.10.8 space of essentially bounded functions, 5.5.9 (5) space of finite-order distributions, 10.10.8 space of functions vanishing at infinity, 5.5.9 (3) space of X-valued p-summable functions, 5.5.9 (6) space of p-summable functions, 5.5.9 (4) space of p-summable sequences, 5.5.9 (4) space of tempered distributions, 10.11.16 space of vanishing sequences, 5.5.9 (3)

342
Spectral Decomposition Lemma, 6.6.6 Spectral Decomposition Theorem, 8.2.12 Spectral Endpoint Theorem, 6.5.5 Spectral Mapping Theorem, 8.2.5 Spectral Purity Theorem, 11.7.11 spectral radius of an operator, 5.6.6 Spectral Theorem, 11.8.6 spectral value of an element of an algebra, 11.2.1 spectral value of an operator, 5.6.13 spectrum, 10.2.7 spectrum of an element of an algebra, 11.2.1 spectrum of an operator, 5.6.13 spherical layer, 6.2.1 -algebra, 6.4.13 -isomorphism, 11.8.3 -linear functional, 2.2.4 -representation, 11.8.3 star-shaped set, 3.1.2 (7) state, 11.9.1 Steklov condition, 6.3.10 Steklov Theorem, 6.3.11 step function, 5.5.9 (6) Stone Theorem, 10.8.10 StoneWeierstrass Theorem for C (Q, C), 11.8.2 StoneWeierstrass Theorem for C (Q, R), 10.8.17 Strict Separation Theorem, 10.4.8 strict subnet, 1.3.5 (2) strictly positive real, 4.1.3 strong order-unit, 5.5.9 (5) strong uniformity, 9.5.5 (6) stronger multinorm, 5.3.1 stronger pretopology, 9.1.2 stronger seminorm, 5.3.3

strongly holomorphic function, 8.1.5 structure of a subdifferential, 10.6.3 subadditive functional, 3.4.7 (4) subcover, 9.6.1 subdifferential, 3.5.1 sublattice, 10.8.2 sublinear functional, 3.4.6 submultiplicative norm, 5.6.1 subnet, 1.3.5 (2) subnet in a broad sense, 1.3.5 (2) subrepresentation, 8.2.2 subspace of a metric space, 4.5.14 subspace of a topological space, 9.2.17 (1) subspace of an ordered vector space, 3.2.6 (2) subspace topology, 9.2.17 (1) SukhomlinovBohnenblustSobczyk Theorem, 3.7.12 sum of a family in the sense of Lp , 5.5.9 (6) sum of germs, 8.1.16 summable family of vectors, 5.5.9 (7) summable function, 5.5.9 (4) superset, 1.3.3 sup-norm, 10.8.1 support function, 10.6.4 support of a distribution, 10.10.5 (6) support of a functional, 10.8.12 support of a measure, 10.9.4 (5) supporting function, 10.6.4 supremum, 1.2.9 symmetric HahnBanach formula, Ex. 3.10 symmetric relation, 1.2.1 symmetric set, 3.1.2 (7) system with integration, 5.5.9 (4) Szankowski Counterexample, 8.3.13

tail filter, 1.3.5 (2) -dual of a locally convex space, 10.2.11 Taylor Series Expansion Theorem, 8.1.9 tempered distribution, 10.11.16 tempered function, 5.1.10 (6), 58; 10.11.6 tempered Radon measure, 10.11.17 (3) test function, 10.10.1 test function space, 10.10.1 theorem on Hilbert isomorphy, 6.3.17 theorem on the equation AX = B , 2.3.13 theorem on the equation XA = B , 2.3.8 theorem on the general form of a distribution, 10.10.14 theorem on the inverse image of a vector topology, 10.1.6 theorem on the repeated Fourier transform, 10.11.13 theorem on the structure of a locally convex topology, 10.2.2 theorem on the structure of a vector topology, 10.1.4 theorem on topologizing by a family of mappings, 9.2.16 TietzeUrysohn Theorem, 10.8.20 topological isomorphism, 9.2.4 topological mapping, 9.2.4 Topological Separation Theorem, 7.5.12 topological space, 9.1.7 topological structure of a convex set, 7.1.1 topological subdifferential, 7.5.8 topological vector space, 10.1.1

343
topologically complemented subspace, 7.4.9 topology, 9.1.7 topology compatible with duality, 10.4.1 topology compatible with vector structure, 10.1.1 topology given by open sets, 9.1.12 topology of a multinormed space, 5.2.8 topology of a uniform space, 9.5.3 topology of the distribution space, 10.10.6 topology of the test function space, 10.10.6 total operator, 2.2.1 total set of functionals, 7.4.11 (2) totally bounded, 4.6.3 transitive relation, 1.2.1 translation, 10.9.4 (1) translation of a distribution, 10.11.17 (7) transpose of an operator, 7.6.2 trivial topology, 9.1.8 (3) truncator, 9.6.19 (1) truncator direction, 10.10.2 (5) truncator set, 10.10.2 twin of a Hilbert space, 6.1.10 (3) twin of a vector space, 2.1.4 (2) Two Norm Principle, 7.4.16 two-sided ideal, 8.3.3, 132; 11.6.2 Tychonoff cube, 9.2.17 (2) Tychonoff product, 9.2.17 (2) Tychonoff space, 9.3.15 Tychonoff Theorem, 9.4.8 Tychonoff topology, 9.2.17 (2) Tychonoff uniformity, 9.5.5 (4) T1 -space, 9.3.2 T1 -topology, 9.3.2 T2 -space, 9.3.5 T3 -space, 9.3.9 T31 /2 -space, 9.3.15

344
T4 -space, 9.3.11 ultrafilter, 1.3.9 ultrametric inequality, 9.5.14 ultranet, 9.4.4 unconditionally summable family of vectors, 5.5.9 (7) unconditionally summable sequence, 5.5.9 (7) underlying set, 2.1.3 Uniform Boundedness Principle, 7.2.5 uniform convergence, 7.2.10, 105; 9.5.5 (6) uniform space, 9.5.1 uniformity, 9.5.1 uniformity of a multinormed space, 5.2.4 uniformity of a seminormed space, 5.2.2 uniformity of a topological vector space, 10.1.10 uniformity of the empty set, 9.5.1 uniformizable space, 9.5.4 uniformly continuous mapping, 4.2.5 unit, 10.9.4 unit ball, 5.2.11 unit circle, 8.1.3 unit disk, 8.1.3 unit element, 11.1.1 unit sphere, Ex. 10.6 unit vector, 6.3.5 unital algebra, 11.1.1 unitary element, 11.7.1 unitary operator, 6.3.17 unitization, 11.1.2 unity, 11.1.1 unity of a group, 10.9.4 (1) unity of an algebra, 11.1.1 unordered sum, 5.5.9 (7) unorderly summable sequence, 5.5.9 (7) Unremovable Spectral Boundary Theorem, 11.2.6

upper bound, 1.2.4 upper envelope, 3.4.8 (3) upper right Dini derivative, 4.7.7 upward-filtered set, 1.2.15 Urysohn Great Lemma, 9.3.13 Urysohn Little Lemma, 9.3.10 Urysohn Theorem, 9.3.14 2-Ultrametric Lemma, 9.5.15 vague topology, 10.9.5 value of a germ at a point, 8.1.21 van der Waerden function, 4.7.7 vector, 2.1.3 vector addition, 2.1.3 vector field, 5.5.9 (6) vector lattice, 3.2.7 vector space, 2.1.3 vector sublattice, 10.8.4 (4) vector topology, 10.1.1 Volterra operator, Ex. 5.12 von NeumannJordan Theorem, 6.1.9 V -net, 4.6.2 V -small, 4.5. weak derivative, 10.10.5 (4) weak multinorm, 5.1.10 (4) weak topology, 10.3.5 weak topology, 10.3.11 weak uniformity, 9.5.5 (6) weaker pretopology, 9.1.2 weakly holomorphic function, 8.1.5 weakly operator holomorphic function, 8.1.5 Weierstrass function, 4.7.7 Weierstrass Theorem, 4.4.5, 46; 9.4.5 Well-Posedness Principle, 7.4.6 Wendel Theorem, 10.9.4 (7) Weyl Criterion, 6.5.4 X-valued function, 5.5.9 (6) Young inequality, 5.5.9 (4) zero of a vector space, 2.1.4 (3)

10.11.4 5.6.2 5.6.3 10.9.4 (7) 6.4.13 8.3.5 5.6.3 11.6.11 8.1.18 Ё 10.11.11 C - 6.4.13 8.5.6 7.6.8 2.2.9 (5) 6.3.8 1.3.1 10.5.5, 10.5.7 10.10.9 (1) 10.3.1 - 10.3.1 - 10.3.5 - 10.3.1 3.7.1 5.1.10 (8) 4.5.11 4.1.13 4.1.13 4.7.4 - 3.1.1 - 3.1.11 - 3.1.6

- 11.9.11 3.8.9 4.8.2 7.4.1 1.2.4 1.2.9 4.1.13 1.2.4 1.2.9 10.11.2 - 10.9.4 (1) 10.11.2 10.3.3 2.3.15, 7.4.16 2.3.3 7.6.5 6.4.8 4.5.3 2.1.7 6.2.5 7.4.9 10.3.3 4.8.2 11.1.1 10.10.7 (5) 6.1.8 4.1.13 10.9.4 (3) 10.2.7 11.4.1 11.6.2 11.4.5 8.3.3

346
11.4.5 4.5.11 2.2.5 6.3.17 7.4.6 - 11.8.3 6.4.13 8.5.1 5.5.9 (4), 10.9.3 5.5.9 (6) 8.1.20 10.9.4 (1) 10.9.3 8.2.1 11.3.1 10.9.4 (1) - 10.3.1 - 10.3.5 - 10.3.1 9.4.17 9.4.21 4.8.5 9.4.22 3.7.4 8.4.8 1.1.4 9.5.12 3.1.2 (9) 3.1.2 (4) 3.2.4 3.2.5 3.2.4 2.3.1 2.3.1 10.5.3 4.4.7, 9.4.4 6.5.4 8.3.11 10.7.1 4.5.6 7.4.20 5.4.5 5.4.2 7.5.1 Ё 8.5.14 8.5.22 6.2.14

8.4.2 4.6.7 9.2.17 (2) 10.8.16 9.4.18 10.8.7 11.9.7 1.2.20 9.6.3 2- 9.5.15 7.6.6 3.8.2 3.6.4 9.3.12 7.6.11 6.6.6 2.3.16 3.8.3 3.7.9 3.2.15 7.1.1 11.9.3 7.3.4 - 8.4.1 6.2.8 9.3.10 9.3.13 10.9.3 10.9.4 (3) 10.9.4 (3) 10.9.4 (1) 10.9.4 (2) 10.9.4 (1) 10.9.4 (3) 10.9.4 (2) 10.9.1 10.11.17 (3) 10.9.4 (1) 10.9.4 (3) F- 10.9.3 4.1.1 Ё 4.6.8 3.1.2 (5) 4.7.1 3.1.2 (8) 3.1.2 (6) 7.1.3

1.2.19 9.1.4, 4.1.11 9.4.2, 4.4.1 4.4.4 3.6.1 3.8.1 3.1.2 (2) 2.2.9 (5) 1.2.19 4.5.3 1.2.15 4.7.1 4.7.1 8.1.1 8.1.1 5.4.3 4.6.3 6.3.1 4.7.4 9.1.4, 4.1.11 4.7.1 4.5.10 3.4.9 1.2.2 4.2.8 10.8.9 4.7.1 8.2.9 4.7.1 1.2.2 3.1.2 (7) 1.2.15 2.1.1 3.2.12 2.3.1 8.2.2 9.5.9 5.1.6 8.3.8 5.1.10 (2) 5.1.10 (4) 5.3.9 5.1.8 3.4.2 1.2.15 10.10.2 (3) 6.3.7 Ё 5.5.9 (4) 3.4.5 6.1.5

347
5.5.9 (4) 5.1.10 (7) 4.1.1 (3), 9.5.7 (3) 5.1.9 5.1.10 (8) 5.6.1 5.1.10 (7) 10.8.12, 10.9.4 (5) 10.10.5 (6) 9.6.4 1.1.2 1.1.2 3.4.2 1.1.1 1.1.1 3.1.14 3.1.14 7.4.19 1.1.3 (5) 9.2.12 1.3.5 (1) 11.9.1 3.7.2 9.3.7 4.1.9, 9.1.1 4.1.5 2.2.1 3.1.7 2.3.5 (5) 2.2.9 (4) 6.6.1 8.3.6 2.2.1 2.2.1 8.2.2 Ё 8.5.1 7.6.9 5.6.10 10.9.4 (4) 5.1.10 (7) 3.2.6 (3) 8.5.9 8.5.9 3.2.6 (3) 6.5.1 10.9.4 (1) 7.6.2 6.4.5 6.3.17 8.5.2

348
6.5.1 6.3.14 6.2.7 6.2.12 1.1.3 (2) 1.2.1 1.2.2 1.2.2 3.2.1 1.1.3 (3) 1.2.1 1.2.1 1.1.3 (3) 1.2.2 1.1.3 (3) 1.2.3 (5) 1.2.3 (4) 9.2.4, 4.2.2 4.2.5 10.10.5 (9) 4.8.2 11.1.5 C - 11.7.8 4.4.2 2.1.4 (3) 3.2.6 (2) 3.3.2 9.2.17 (1) 1.3.5 (2) 9.6.1 9.6.2 4.4.2 9.6.2 9.5.7 3.7.6 2.1.2 7.6.8 10.5.1 10.5.1 4.5.13 1.2.2 1.2.3 (2) 10.10.5 (3) 1.2.16 9.6.15 2.3.5 (6)

2.3.5 (5) 2.3.5 (1) 1.2.16 2.3.4 4.5.2 4.1.16 4.1.17 5.5.9 (4) 9.1.1 1.2.2 1.2.3 (2) 10.9.4 (4) 8.2.2 11.1.7 8.2.2 8.2.2 - 11.8.3 9.1.1 11.6.8 10.11.3 6.3.16 10.11.15 10.11.19 7.5.5 7.1.5 7.4.17 7.4.10 7.3.5 7.4.6 10.9.10 10.10.11 7.5.11 7.4.18 7.3.13 7.2.5 7.2.4 7.2.12 7.2.11 7.6.7 7.6.13 6.4.9 6.4.12 11.1.2 2.2.9 (4) 2.2.9 (3) 8.2.11

6.2.3 2.1.4 (4) 9.5.5 (4) 6.1.4 9.2.17 (2) 9.3.2, 9.2.17 (2) 10.10.5 (4) 10.10.5 (4) 5.1.4 1.2.3 (3) 9.5.5 (3) 9.2.9 4.8.8 5.5.1 5.5.9 (5) 10.10.9 (3) 7.1.8 4.7.2 2.1.3 3.2.2 6.1.7 6.1.10 (4) 2.1.4 (2) 3.2.8 9.4.4 5.5.9 (5) 10.2.9 11.6.7 4.1.1 4.5.5 10.10.9 (2) 9.5.10 9.5.9 5.1.6 10.2.7 5.4.1 5.2.13 5.1.9 5.1.10 (8) 5.1.10 (8) 5.4.1 9.6.9 5.1.5 6.1.7 6.1.10 (3) 9.1.1 9.5.1 6.3.14

349
10.2.11 8.3.10 5.4.1 9.1.7 10.1.1 9.3.15 10.1.3 9.4.20 9.3.11 9.3.2 9.3.9 9.3.15 9.3.5 11.6.5 5.5.2 10.11.16 10.11.6 K - 3.2.8 10.9.11 6.3.16, 10.11.12 9.5.1 4.1.5 9.5.9 5.2.4 5.2.2 9.5.5 (6) 9.5.5 (6) 9.5.5 (6) 9.5.5 (4) 10.1.10 11.6.11 5.6.16 9.6.6 10.10.4 10.10.5 (3) 10.11.16 10.11.17 (7) 10.10.5 (2) 10.10.5 (1) 10.10.5 (9) 10.11.16

350
10.10.5 (5) 8.5.9 8.5.9 5.6.13 11.2.1 1.2.12 3.2.7 1.2.13 5.6.9 (1) Ё 10.9.4 (7) 10.10.5 (9) 9.6.17 10.9.4 (7) 10.10.5 (9) 1.1.3 (4) 5.5.9 (7) 5.5.9 (7) 5.5.9 (7) 1.2.16 4.5.2 4.5.2 V - 4.6.2 - 8.3.2 5.5.9 (4) 1.2.3 (4) 6.6.3 1.1.1 3.1.7 7.3.8 7.3.3 2.2.1 1.1.3 (1) 1.1.3 (3) 11.9.1 5.6.13 11.2.1 5.6.6 9.6.19 (1) 3.5.1 3.7.8 7.5.8 1.1.3 (5) 2.1.4 (5) 6.1.10 (5) 5.5.9 (7) p 5.5.9 (6) 5.5.9 (7)

5.5.9 (4) 10.6.7 4.6.10 8.5.18 7.4.4 7.4.7 7.4.5 5.6.12 7.2.9 9.2.2, 4.1.19 4.7.6 4.4.5, 9.4.5 10.9.9 10.9.4 (7) 7.2.2 8.2.3 11.3.2 11.2.3 11.9.12 11.8.4 6.6.7 8.3.9 8.2.7 8.1.3 10.11.2 5.5.9 (7) 10.7.5 7.2.10 10.8.6 4.8.3 7.4.11 (2) 8.3.4 4.4.9 3.3.4 8.1.7 10.6.5 3.6.5 3.3.8 11.4.8 6.2.2 7.4.11 10.10.5 (9) 8.1.10

10.4.9 10.4.6 10.4.5 3.8.11 10.5.8 6.5.5 10.3.9 8.5.20 10.9.10 10.10.11 11.5.3 11.5.1 11.8.6 10.11.9 11.7.9 11.6.9 9.2.8 10.1.6 8.2.12 9.6.20 8.2.13 8.1.9 AX = B 2.3.13 XA = B 2.3.8 C - 11.9.10 5.6.22 10.1.4 10.2.2 10.6.3 5.3.2 6.3.3 5.6.9 3.8.7 10.5.9 11.6.6 9.2.11 6.6.9

351
10.10.13 10.11.18 8.5.21 6.2.10 8.2.5 10.11.12 4.7.5 3.8.11 7.5.12 10.4.8 3.8.14 10.7.4 6.3.2 10.11.14 10.9.4 (3) 10.11.5 (3) 5.5.9 (7) 5.3.5 6.4.1 3.2.16 5.5.9 (4) 6.3.16 8.4.8 6.5.2 11.8.6 XA = B 7.4.12 6.3.11 10.8.10 10.8.17 C (Q, C) 11.8.2 3.7.11 10.8.20 9.4.8 10.11.5 (6) 9.3.14 AX = B 7.4.14 6.1.9 8.5.8 10.9.4 (6) 10.10.5 (8) 3.5.3 3.5.4 3.8.12

352
3.5.3 7.5.9 3.7.13 7.5.10 7.6.12 4.5.12 8.4.6 10.10.10 11.2.4 5.6.19 6.1.3 8.5.17 9.1.7 9.1.8 (3) 10.1.11 9.1.8 (4) 10.9.6 10.1.3 10.2.1 10.4.4 4.1.9 5.2.8 9.5.5 (6) 10.10.6 10.10.6 10.11.6 10.11.6 9.5.3 9.5.5 (6) 10.3.5 10.4.1 10.9.5 T1 9.3.2 T2 9.3.5 T3 9.3.9 T31 /2 9.3.15 T4 9.3.11 4.1.13 4.1.13 3.4.11 4.1.13 3.6.1 4.1.13 9.4.1

9.5.13 1.3.9 6.3.10 - 11.4.3 - 1.2.3 (4) - 5.3.11 - 5.1.10 (5) - 2.1.4 (6) 1.3.3 4.5.2 1.3.5 (2) 6.1.2 6.1.4 6.1.2 6.1.1 Ё 8.1.12 5.6.8 3.1.13 3.5.5 3.7.10 10.9.4 (4) 2.2.4 - 2.2.4 3.8.6 3.4.7 (2) 3.2.6 (3) 3.4.7 (4) 3.4.6 8.2, 11.3 11.8.7 3.1.7 10.11.6 3.4.4 9.6.13 8.1.4 3.4.8 (2) 5.5.9 (4) 9.6.17 10.10.4 10.10.5 (3) 10.11.16 10.11.17 (7) 10.10.5 (2) 10.10.5 (1) 10.10.5 (9) 10.6.4

10.10.1 4.3.3 10.10.1 5.5.9 (6) 9.6.4 9.6.19 (1) 10.11.6 9.6.4 10.10.5 (4) 9.6.4 11.6.4 10.11.1 1.2.19 4.1.3 2.2.9 (4) 3.2.12 3.2.12 3.6.3 (4) 4.1.3 5.2.11 5.1.10 (8) 7.6.5 7.6.3 10.2.13 6.4.1 3.3.6 11.1.1 11.1.3 1.2.10 1.2.10 1.2.6 1.2.6 11.7.1 11.1.5 6.2.5 3.2.5, 11.9.4 11.1.3 11.7.1 11.7.1 5.6.4, 8.2.1 2.3.1 3.4.11 2.3.1 5.1.1 (3) 9.6.14

353

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