S.S. Kutateladze
. . .
C. C.
A 4- ,
2001
517.98 22.1695
. . . 4- ., . : - - , 2001. xii+354 c. ( ).ISBN 5861341036. . ,
, , , , . - . , ..: 347. . . . . 16020800001082(03)2001 .
ISBN 5861341036
c . ., 2001
c
. . . , 2001
viii
xii
1. 1.1.
.....................................
1.2.
3
.........................................
7
............................................
10
2. 2.1. 2.2.
12................
12
.............................
15
2.3.
1
.......................
1.3.
1
.........................
18
............................................
24
3.
26
3.1.
..........
26
........
29
3.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . .
32
3.4.
35
3.5.
38
3.2.
.........................
iv
3.6. . . . . . . . . . . . . . . . . . . . . . . . . . .
41
3.7.
.........
44
3.8.
........
46
............................................
51
4.
53
4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . .
53
4.2.
...
56
..............................
59
....................................
60
..........................................
62
4.3. 4.4. 4.5.
4.6.
.........................
65
4.7.
.........................
68
4.8.
...........
71
............................................
72
5. 5.1.
74
......................
74
5.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . .
79
5.3.
82
...........................
5.4. 5.5.
.....
85
..........................
87
5.6.
...............
97
............................................
104
v
6.
106
6.1. 6.2. 6.3.
....
106
..................................
111
................................
114
6.4. 6.5.
................
119
............................
122
6.6.
...............
125
............................................
129
7.
131
7.1.
.......................
131
7.2.
.......................
134
7.3.
...............
7.4.
.
141
........
147
.........................
150
............................................
155
7.5. 7.6.
138
8. 8.1.
158.
158
.......
165
8.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . .
172
8.4.
175
8.2.
........................
8.5.
...........
179
............................................
187
vi
9. 9.1. 9.2.
......................
190
...................................
193
9.3. 9.4.
190
..............
196
....................................
201
9.5.
207
9.6.
.................
213
............................................
218
10. 10.1.
220
...........................
10.2.
.................
10.3.
........
10.4. , 10.5.
220223226
..
228
.........................................
230
10.6.
.......
232
....................
234
..........................
236
...................................
243
10.7. 10.8. C(Q, R) 10.9.
10.10. D D 0
..........................
251
10.11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . .
260
272
............................................
vii
11.
274
11.1.
......
274
.......................
276
11.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
278
11.4.
.............
280
11.5. C(Q, C)
......................
281
11.6.
.....................
283
...................
288
11.2.
11.7. C -
11.8.
290
11.9. - C -
.......
294
............................................
300
303
323
327
345
, . . , . , . , . , . , . . . , , (, ). . , , , . , . , . , , ,
. , . , , . , , , , , . , . . , . , . , . , . . -
ix
. . . , . , . ., , . . . . , 1900 ., . . . , . , . , . , . , .
., . , , . . . . , . . . , (, . , , , , ). , , , : , . , , , , . :
, . , . , , , .
x
, (. 1449). . , . . . . . , . . , . . , . ., . . , . . , . . . .
. . 1948 . . . , , , 1974 ., . . , , . . , , , . .
. . , , . , . , . . . . , , .
. , , Kluwer Academic Publishers 1996 ., , , .
, , .C.
1
1.1. 1.1.1. . A B F A B. F A B , , A B.1.1.2. . F A 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 A B, F 1 := {(b, a) B A : (a, b) F }
B A, F . , F F 1 .(2) F A F A A.(3) F A 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 B A B, F dom F = A. IU A = U . IU . F A B F : A B. ,
dom F = A im F B. im F = B : F A B., F 1 B A , F : A B .(4) ., F :
A B B (ba )aA , a 7 ba (a A), (ba ). , (a, b) F , b = ba . , .(5) F
A B U A. F (U B) U 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 B G C D G F := {(a, d) A 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 imF . , F F 1 =
I imF , F |dom F . CB1.1.7. F A B, G B C U A. G F A C G F (U ) =
G(F (U )). CB1.1.8. F A B, G B C, H C D. H (G F ) A D (H G) F A D .
CB1.1.9. . 1.1.8 H G F .1.1.10. F, G, H . H GF =
[
F 1 (b) H(c).
(b,c)G
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) B1.1.11. . 1.1.10 , , , ( , 1.1.1). . ( ) , ,
.1.1.12. G F GF =
[
F 1 (b) G(b).
b imF
C 1.1.10 : H := G, G := I imF F := F . B
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 X0 .(2) () X, 1 () X. 1 ().(3) f : X Y Y . X : f 1 f .
1.1.10[f 1 f =f 1 (y1 ) f 1 (y2 ).(y1 ,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 =
[
1 (x) 1 (x).
xX
f : X Y . f f X, . . f : X Y , f = f , f 1 f . CB(5) (X, ) (Y, )
. f : X Y (. . x y f (x) f (y)) , f 1 f . CB1.2.4. . (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}.
CB1.2.8. . 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. , . CB1.2.15. . (X,
) , X 2 = 1 , . -
1.3.
7
. , , .1.2.16. . X () X. ( ) N X () . ( , N := {1, 2, 3 . . .
}.)1.2.17. , . CB1.2.18. . 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 6= , 6 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 F F B . .1.3.4. F 2X , (1) F 6= , 6 F ;(2) A F , A B X B F ;(3)
A1 , A2 F A1 A2 F . CB1.3.5. .(1) F X Y B 2X . F (B) := {F (B) :B
B}. , F (B) . , F (B) := fil F (B). F X B dom F 6= 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 ). . CB
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) . CB1.3.8. N F (X). N F0 := sup N . F0 = {F : F N
}.C , F0 . , / F0 , N , F0 6= . 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 . B1.3.9. . F
(X) X .1.3.10. .C 1.3.8 , , . 1.2.20. B1.3.11. F , A X A F , X \ A
F .C : A 6 F B := X \ A 6 F . , A 6= B 6= . F1 := {C 2X : A C F }.
A 6 F 6 F1 B F1 F1 6= . 1.3.4 (2) 1.3.4 (3). , F1 . F1 F . F , F1 =
F . : B 6 F B F.: F1 F (X) F1 F . A F1 A 6 F , X \A F . X \A F1 , .
. = A(X \A) F1 , . B
10
. 1.
1.3.12. f X Y F X, f (F ) Y . CB1.3.13. X := XF0 := {F F (X) : F
F0 } F0 F (X). X .C , 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 6= 6 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 . B1.1. , - - .1.2. [0, 1]
[0, 1]? [0, 2]?1.3. : , , RM RN M, N .1.4. R, S, T :(R S)1 = R1 S 1
;
(R S)1 = R1 S 1 ;
(R S) T = (R T ) (S T );
R (S T ) = (R S) (R T );
(R S) T (R T ) (S T );
R (S T ) (R S) (R T ).
1.5. X X 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 Z) M (X, Y ) M (Y, Z);(2) M
(X 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} =6 . R
. R R?
2
2.1. 2.1.1. . , , . X A (X, +) A X, : AX 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, +, ). , , , X .2.1.4. .(1) F
F.(2) (X, F, +, ) . -
2.1.
13
(X, F, +, ), : (, x) 7 x s F x X, . X X . F := R X X.(3) (X0 ,
F, +, ) (X, F, +, ), X0 X X0 F X0 X. X0 X. , , X0 X. , X X 0. X , ,
, .(4) (X ) Q F. , , X := X , . . x : X , x := x() X ( , 6= ). X
:(x1 + x2 )() := x1 () + x2 () (x1 , x2 X , );( x)() := x() (x X ,
F, )(, , x : x x). X F (X ) . := {1, 2, . . . , N } X1 X2 . . . XN
:= X . , X = X , X := X . := {1, 2, . . . , N }, X N := X .(5) (X )
P F. Q 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 (x2 )) (x1 , x2 X );x := (1 (x)) (x X , F)., , S1 , S2 X, F F
, S1 + S2 := +{S1 S2 };S1 := ( S1 );
S1 := {}S1 .
X F. - X X0 X/X0 .2.1.5. X Lat(X) X . Lat(X) .C , inf Lat(X) = 0
sup Lat(X) = X. , . 1.2.17, . B2.1.6. . X1 , X2 Lat(X) X1 X2 = X1 +
X2 . , E Lat(X) inf E = {X0 : X0 E }. E , sup E = {X0 : X0 E }.
CB
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. .C 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 x2 + x = x1 ,
x X1 + X2 , , = 0. x1 = x2 = 0, X1 X2 = 0. , X2 +{x : F} = X2 X2 .
, x = 0. x 6= 0. X1 X2 = X1 + X2 = X. B2.2. 2.2.1. . X, Y F. T X Y
, T X Y . T : X Y , , ( , ). T S X Y dom S 6= 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). CB
16
. 2.
2.2.3. L (X, Y ) X Y Y X . CB2.2.4. . 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 . CB2.2.8. . X, Y, Z ,
T L (X, Y ) S L (Y, Z). , S T L (X, Z). S T ST . , (S, T ) 7 ST , ,
: L (Y, Z) L (X, Y ) L (X, Z). , E L (Y, Z), T L (X, Y ), E T := (E
{T }).2.2.9. .(1) T , T 1 .(2) X1 X X2 , X2 X/X1 ., : X X/X1 ,
2.2.
17
X2 , . . x2 7 (x2 ), x2 X2 , . CBQ(3) X := X (X ) . Pr : X X ,
Pr x := x , (= ). , Pr : Pr L (X , X ). , L (X ) := L (XQ, X ), X X
, X := X , X := 0 6= X := X .(4) X := X1 X2 . +1 X X1 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 . , 2PX1 + PX2 = IX . , PX:=
PX1 PX1 = PX1 , . . 1 . , P L (X) P (X) P 1 (0). T L (X), PX1 T PX1
= T PX1 , T (X1 ) X1 , . . X1 T . CB 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
T1 0T .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 X2 X1 X2 ,
x1x2
7
T10
0T2
x1x2
.
T L (X1 X2 , Y1 Y2 ). CB(5) E X P, 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. , . CB
2.3.
19
2.3.3. . . , ., 1- Y X@2 43@R ?@V - W5 , 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 .T
S
2.3.4. . X Y Z ( Y ) , ker S = im T . . . . Xk1 Xk Xk+1 . . . Xk
, Xk1 Xk Xk+1( ). , ( , , , ).2.3.5. .T
S
(1) X Y Z ,. . ST = 0. .T
(2) 0 X Y , T . ( 0 X , , L (0, X) (. 2.1.4 (3)).)T
(3) X Y 0 , T . (, Y 0 L (Y, 0).)(4) T L (X, Y ) T , 0 X Y 0
.
20
. 2.
(5) X0 X. : X0 X () : x0 := x0 x0 X0 . X/X0 - :X X/X0 .
0 X0 X X/X0 0 . ( , , .) . , , , T
S
0XY Z0 , . Y0 := im T , , , , :0
- X
0
?- Y0
T
- Y
S
- Z??- Y- Y /Y0
- 0
- 0
, , - . CB(6) T L (X, Y ) . T
0 ker T X Y coker T 0, T .2.3.6. . T T0( T T0 ), X0
-X@TT0@@R ?Y
. . T0 = T , : X0 X .
2.3.
21
2.3.7. X, Y X0 X. T0 L (X0 , Y ) T L (X, Y ).C T := T0 PX0 , PX0
X0 . B2.3.8. XA = B. X, Y, Z ; A L (X, Y ), B L (X, Z). X
A- Y@XB@@R ?Z
X L (Y, Z) , ker A ker B.C : , B = X A ker A ker B, .: X := B A1
. , x X X A(x) = B (A1 A)x = B(x + ker A) = Bx. , X0 :=X |im A . X
. y im A z1 , z2 X (y). z1 = Bx1 , z2 =Bx2 , Ax1 = Ax2 = y. B(x1 x2
) = 0. , z1 = z2 . 2.3.7, - X X0 Y . B2.3.9. . 2.3.8 A , X .
CB2.3.10. .C 2.3.8 2.3.9. B2.3.11. T () , T , . . :
22
. 2. coim T 6X
T im TT
?- Y
T . CB2.3.12. X f0 , f1 , . . . , fN X # . f0 f1 , . . . , fN ,
ker f0 Nj=1 ker fj .C (f1 , . . . , fN ) : X FN , (f1 , . . . , fN
)x := (f1 (x), . . . , fN (x))., ker(f1 , . . . , fN ) = Nj=1 ker
fj . 2.3.8 X
(f1 ,...,fN )
- FN
@@f0@@R ?@F FN # , . B2.3.13. AX = B. X, Y, Z ; A L (Y, X), B L
(Z, X). AX
Y
6@I@XB @Z X L (Z, Y ) , im A im B.
2.3.
23
C : im B = B(Z) = A(X (Z)) A(Y ) = im A.: Y0 ker A Y A0 := A|Y0
. A0 Y0 im A. X := A10 B, , . B2.3.14. . 2.3.13 A , X . CB2.3.15. .
2.3.8 2.3.13 . , .2.3.16. . S L (Y, Z) T L (X, Y ). , , 1 , . . . ,
6 , :
()
1230 ker T ker ST ker S
345coker T coker ST coker S 0
. CB
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# 7 hx | x# i
(x X, x# X # ).
, X X ## .2.8. , , . .##
(X) = X ## dim X < +.
2.9. ?2.10. ?2.11. T , T n1 6= 0 T n = 0 - n. , T 0 , T, . . . ,
T n1 .2.12. , .2.13. XA = B AX = B ( X ).2.14. ?2.15. , .
25
2.16. (xe )eE (x#e )eE , :hxe | x#e i=1hxe | x#e i=0
(e E );
(e, e0 E, e 6= e0 ).
2.17. (x#e )eE (xe )eE , :hxe | x#e i=1hxe | x#e i=0
(e E );
(e, e0 E, e 6= e0 ).
2.18. .2.19. T
W X Y ZT
W X Y Z , , . , ker = T (ker ) T 1 (im ) = im .
3
3.1. 3.1.1. . F 2 , U . U ( U ( )), (1 , 2 ) 1 U + 2 U U.3.1.2.
.(1) (). ( , () .)(2) := F 2 - .(3) := R2 , - X X.(4) := R2+ , - .
, K , K + K K K K R+ . R2+ \ 0- () , R+ 0- . ( R+ := {t R : t
0}.)(5) := {(1 , 2 ) F 2 : 1 + 2 = 1}. - . X0
3.1.
27
X x X, x + X0 := {x} + X0 X. , L X x L, L x := L + {x} X. CB(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}, - . , . CB(10) F 2 X F X ( ). ,
3.1.2(1)3.1.2 (9) -.3.1.3. X E - X. {U : U im E } ( ). , , im E (
), {U : U im E } ( ). CB3.1.4. . 3.1.3, , , - , , .3.1.5. X Y , U X
V Y -. U V ( ).C U V , U 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 V .
B3.1.6. . X, Y T X Y . F 2 . T ( ), T -.
28
. 3.
3.1.7. . - ( ) , . , . . : , (. 3.4.2).3.1.8. T X Y 1 -, U X 2
-. 2 1 , T (U ) (2 ).C 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 ). B3.1.9. - -.C F X V G W Y F, G ( ). (x1 , y1 ) G F
( v1 )
(x1 , v1 ) F & (v1 , y1 ) G;
(x2 , y2 ) G F ( v2 )
(x2 , v2 ) F & (v2 , y2 ) G.
1 , 2 , (1 , 2 ) , , . B3.1.10. U , V X U , V ( ) F 2 , , F U +
V ( ).C 3.1.5, 3.1.8 3.1.9. B3.1.11. . 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 ).
CB
3.2.
29
3.1.13. :H (U ) = {H (U0 ) : U0 U, U0 }.C 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 ). B3.1.14. . , - - . , ( ) -. , := {(1 , 2
) R2+ : 1 + 2 = 1} H (U ) co(U ). HF 2 (U ) L (U ) lin(U ), U 6= ,
, L () := 0. L (U ) U ( L (X) X). , . . , , . .(N)Xco({x1 , . . . ,
xN }) =k xk : k 0, 1 + . . . + N = 1 . CBk=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.
C (0) 3.1.3. , (x, y) = (x, x) + (0, y x) (x, y) y x (0).
B3.2.3. K X. := {(x, y) X 2 : y x K}. , , K (0). , , K (K) = 0.C ,
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. B3.2.4. . 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- .C x U . x U . , 3.2.8 sup(U ). x sup(U ). , sup(U ) =
inf U . B3.2.10. K- U V sup(U + V ) = sup U + sup V.C , 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. B3.2.11. . 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.C x + y x y = x + y + (x) (y) = y
x B3.2.14. x = x+ x ; |x| = x+ + x .C 3.2.13 y := 0. , |x| = x (x)
= x + (2x) 0 = x + 2x+ = (x+ x ) + 2x+ =x+ + x . B3.2.15. . x, y X
[0, x + y] = [0, x] + [0, y].( , [u, v] := (u) 1 (v) () .)C [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.
B3.2.16. . X , Y K-. Lr (X, Y ) L+ (X, Y ) K-. CB3.3. 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 . CB3.3.4. . X , X0 X Y K-.
T0 L+ (X0 , Y ) T L+ (X, Y ).C 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 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 . B3.3.5. . Y := R 3.3.4 .3.3.6. . x , [0, x] = [0,
1]x.3.3.7. (X, X+ ) , X = X+ X+ .C T X := X+ X+ . f X # . , ker f X
f = 0. T + f [0, T ], . . [0, 1] T + f = T . T |X = 0, 2T [0, T ].
T = 0 f = 0. T (x0 ) 6= 0 - x0 X , = 1 f = 0. B3.3.8. . X , X X T0
X0 . T X, T0 .C 3.3.4. I. T . , T 0 [0, T ] [0, 1] x0 X0 T 0 (x0 )
= T (x0 ) (T T 0 )(x0 ) = (1 )T (x0 ). :T 0 (x) inf{T 0 (x0 ) : x0
x, x0 X0 } = T (x);(T T 0 )(x) inf{(T T 0 )(x0 ) : x0 x, x0 X0 } =
(1 )T (x). , T 0 = 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 0 (dom S)# , 0 S 0 (x0 ) S(x0 ) x0 (dom
S)+ . S(x0 ) = 0 x0 , S 0 = 0S, . S(x0 ) 6= 0 x0 (dom S)+ , S0 E0
S0 (x0 ) = S(x0 ). S0 : S 0 (x0 ) = S(x0 ) x0 dom S0 . = S 0 (x0
)/S(x0 ), . . S0 . E0 , : S 0 = S. B3.4. 3.4.1. . R R +. (+) := + (
R+ ), + + x := x +(+) := + (x R ).3.4.2. . f : X R . epi f := {(x,
t) X R : t f (x)} f , dom f := {x X : f (x) < +} f .3.4.3. . dom
f . , f : X R f X 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.C : 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 . B3.4.6. . p : X R , epi p .3.4.7. dom p 6= 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. CB3.4.8. .(1) , .(2) U X.
0, x U,(U )(x) :=+, x 6 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 Y , im T = Y . U ( X), T (U ) ( Y ).C Y = T (X) = T (nN nU ) =
nN T (nU ) = nN nT (U ) B3.4.11. . U X. x U core U U ( U ), U x
X.3.4.12. f : X R x core dom f . h X f 0 (x)(h) := lim0
f (x + h) f (x)f (x + h) f (x)= inf.>0
f 0 (x) : h 7 f 0 (x)h f 0 (x) : X R.C () := f (x + h). 3.4.8
(4) : R R . 0 core dom . 7 (() (0))/ ( > 0) , . . 0 (0)(1). f 0
(x)(h) = 0 (0)(1).
38
. 3. > 0 h H f 0 (x)(h) = inf= inf
f (x + h) f (x)=
f (x + h) f (x)= f 0 (x)(h).
, h1 , h2 X
f x + 21 (h1 + h2 ) f (x)0=f (x)(h1 + h2 ) = 2 lim0
f 21 (x + h1 ) + 21 (x + h2 ) f (x)= 2 lim0 lim0
f (x + h1 ) f (x)f (x + h2 ) f (x)+ lim=0= f 0 (x)(h1 ) + f 0
(x)(h2 ).
3.4.7 . B3.5. 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 0 (x)). CB3.5.3. . T L
(X, Y ) , f : Y R , x X, T x core dom f . x (f T ) = T x (f ) T.C
3.4.10 , x core dom f . 3.5.2 (4), x (f T ) = ((f T )0 (x)). , h
X(f T )0 (x)(h) = lim0
= lim0
(f T )(x + h) (f T )(x)=
f (T x + T h) f (T x)= f 0 (T x)(T h).
p := f 0 (T x). 3.5.2 (4) , , 3.4.12, p , :(p) = (f 0 (T x)) = T
x (f );(p T ) = ((f T )0 (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 R Y := Y R T0 : (y0 , t) 7t 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).
B3.5.4. . 3.5.3 , . , (p ) (p) : , , , .3.5.5. . X , X0 X p : X R .
() :(p + (X0 )) = (p) + ((X0 )).C . 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 )). B
3.6.
41
3.5.6. . f : X R x core dom f . x (f ) 6= .C p := f 0 (x), : 0 X
. , 0 (p),. . (p ) 6= . 3.5.3, (p) 6= ( = (p) =(p )). 3.5.2 (4).
B3.5.7. . f , f : X R 1
2
x core dom f1 core dom f2 . x (f1 + f2 ) = x (f1 ) + x (f2 ).C
p1 := f10 (x) p2 := f20 (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 ). B3.5.8. .
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 X , seg(x1 , x2 ) := {1 x1 + 2 x2 : 1 , 2 >
0, 1 + 2 = 1}.
42
. 3.
, , V X segV seg V 2 . U , V , 2 V , seg1V (U ) U . . 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 . CB3.6.3. .(1) p : X R x X dom p. x (p)
(p).C , 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). B(2) U V , ,V W .
U W . CB(3) X . x X+ , {x : R+ } X+ .C : 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
6= 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. B(4) U . V U U , U \ V .
3.6.
43
x U , {x} U. CB3.6.4. . p : X R l (p). , ,X := X R, X+ := epi p
Tl : (x, t) 7 t l(x) (x X, t R). l (p) , Tl .C : T 0 X # , T 0 [0,
Tl ].t1 := T 0 (0, 1), l1 (x) := T 0 (x, 0);t2 := (Tl T 0 )(0,
1),
l2 (x) := (Tl T 0 )(x, 0).
, t1 0, t2 0, t1 +t2 = 1; l1 (t1 p), l2 (t2 p) l1 +l2 = l. t1 =
0, l1 = 0, . . T 0 = 0 T 0 [0, 1]Tl . t2 = 0, t1 = 1, . . T 0 = Tl
T 0 [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 0 = t1
T l .: l = 1 l1 +2 l2 , l1 , l2 (p) 1 , 2 > 0, 1 + 2 = 1. T 0 :=
1 Tl1 T 00 := 2 Tl2 , T 0 [0, Tl ], T 0 + T 00 = Tl . , [0, 1], T 0
= Tl . (0, 1), 1 = ., l1 = l. l2 = l. B3.6.5. . p : X R . x X l
ext((p)) , l(x) = p(x).C , . . , p : ext((p)) 6= . X := X R X+ :=
epi p X0 := 0R. , X+ X0 = 0R+ = 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 (p0 (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). B3.6.6. . p1 , p2 : X R . p1 p2 ( RX )
, (p1 ) ext((p2 )).C , p1 p2 (p1 ) (p2 ). , 3.6.5,p2 (x) = sup{l(x)
: l ext((p2 ))}. B3.7. 3.7.1. . (X, F, +, ) F. (X, R, +, |R X ) (X,
F, +, ) XR .3.7.2. . X f X # . Re f : x 7 Re f (x) (x X). Re : (X #
)R (XR )# .3.7.3. Re (X # )R (XR )# .C 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 . B3.7.4. . Re1 : (XR )# (X # )R
.3.7.5. . 3.7.3 Re1 g : x 7 g(x) ig(ix) (g (XR )# , x X). F := R
Re1 .3.7.6. . (X, F, +, ) F. p : X R , dom p 6= 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) = Re1 ((p));
Re (||(p)) = (p).
C 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 := Re1 g. f (x) =
0, |f (x)| p(x). f (x) 6= 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). B
46
. 3.
3.7.10. X , p : X R X0 X. () ||(p + (X0 )) = ||(p) + ||((X0 )).C
3.7.9 3.5.5, :||(p + (X0 )) = Re1 ((p + (X0 ))) = Re1 ((p) + ((X0
))) == Re1 ((p)) + Re1 (((X0 ))) = ||(p) + ||((X0 )). B3.7.11. X, Y
, T L (X, Y ) p : Y R . p T , ||(p T ) = ||(p) T.C 2.3.8 3.7.10,
||(p T ) = ||(p + (im T )) T = (||(p) + ||((im T ))) T == ||(p) T +
||((im T )) T = ||(p) T. B3.7.12. . 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 . CB3.8. 3.8.1. . R (. . R ). X f : X R
, t R {f t} := {x X : f (x) t};
3.8.
47
{f = t} := f 1 (t);{f < t} := {f t} \ {f = t}. {f t}, {f =
t}, {f < t} f . , {f = t} .3.8.2. . T R t 7 Ut (t T ) X. f : X R
, {f < t} Ut {f t} (t T ) , t 7 Ut .C : T s t ( ). s < t, Us
{f s} {f < t} Ut .: f (x) := inf{t T : x Ut }. f : X R. t T {f
< t} , {f < t} Ut . x {f < t}, f (x) < +, s T , x Us s
< t. , {f < t} Us Ut . , x Ut , f f (x) t, . . Ut {f t}.
B3.8.3. , . 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 t, f
(x) r r > t, f (x) t. B3.8.5. X S . t R Ut := , t < 0, Ut :=
tS t 0. t 7 Ut (t R) .C 0 t1 < t2 x t1 S, x (t1 /t2 ) t2 S. ,x
t2 S. B
3.8.
49
3.8.6. . pS : X R , {pS < t} tS {pS t} (t R+ ) {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}.C
S pS . x X. pS (x) 0 . > 0.
tpS (x) = inf{t > 0 : x tS} = inf t > 0 : x S == 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 (
t2t1x1 +x2,t1 x1 + t2 x2 = (t1 + t2 )t1 + t2t1 + t 2 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 ).
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 . B3.8.8. . S X , dom pS = X. , S , pS . p {p < 1} {p 1} .
CB3.8.9. . H X , X/H . X/H X ( H). X , - X. XR X X.3.8.10. X X # .
CB3.8.11. . X , U X L X. L U = , H X , H L H core U = .C , , core U
6= ( ) , , 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) 6 (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
:=Re1 f H := {f = f (x)}. , L H H. , H . B3.8.12. . 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 V , V U . ,
U V V . CB3.1. , , .3.2. , .3.3. , , . . . .3.4. . , ?
52
. 3. S
3.5. S1 S2 S = 01 S1 (1)S2 . , S S1 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 2- , . ? ?3.14. ?3.15. RN
K-?3.16. ?3.17. l .3.18. , .3.19. C X H(C) H(C) = {(x, t) X R : x
tC}. .
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 y. , X .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+ ).C
4.1.2 (1)4.1.2 (3) 4.1.1 (1)4.1.1 (3) . B4.1.3. . (X, d) R+ \ 0. B
:= Bd, := {d }
( ), B := B d, := {d < }
54
. 4.
( ). B (x) x B x.
B (x) x.4.1.4. , , . CB4.1.5. . , (X, d) X 2 , UX , Ud , , , U ,
, . X := UX := {}. UX ().4.1.6. U . (1) U fil {IX };(2) U U U 1 U
;(3) ( U U ) ( V U ) V V U ;(4) {U : U U } = IX . CB4.1.7. . 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). CB4.1.9.
. : x 7 (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. . .
CB4.1.13. . 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 . CB4.1.15. . 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. . , , . . CB4.1.19. U x :(1) x U ;(2) F , F x U F ;(3) (x
) U , x.C (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 . B4.1.20. . 4.1.19 (2) , F , 4.1.19 (3) := N. :
.4.2. 4.2.1. f : X Y X , Y X Y . :
4.2.
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 ).C (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 ). B(1)(2)(3)(4)(5)
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. .C 4.2.1 (5). B4.2.4. f : X Y UX , UY
X Y. :(1) ( V UY ) ( U UX ) ( x, y)(x, y) U (f (x), f (y)) V ;
58
. 4.
(2) ( V UY ) f 1 V f UX ;(3) f (UX ) UY , f : X 2 Y f : (x, y) 7
(f (x), f (y));(4) ( V UY ) f 1 (V ) UX , . . f 1 (UY ) UX .C ,
1.1.10 U X 2 V Y 2[
f 1 V f =
f 1 (v1 ) f 1 (v2 ) =
(v1 ,v2 )V
= {(x, y) X : (f (x), f (y)) V } = f 1 (V );[f U f 1 =f (u1 ) f
(u2 ) =2
(u1 ,u2 )U
= {(f (u1 ), f (u2 )) : (u1 , u2 ) U } = f (U ). B4.2.5. . f : X
Y , ( , ) 4.2.4 (1)4.2.4 (4), ( ) .4.2.6. .C f : X Y , g : Y Z h :=
g f : X Z. , h (x, y) = (h(x), h(y)) = (g(f (x)), g(f (y))) == g (f
(x), f (y)) = g f (x, y) x, y X. , h (UX ) = g (f (UX )) g (UY ) UZ
4.2.4 (3). 4.2.4 (3), , h . B4.2.7. . CB4.2.8. . E X Y UX , UY . E
() , ( V UY )
\f E
f 1 V f UX .
4.3.
59
4.2.9. . . CB4.3. 4.3.1. (X1 , d1 ) (X2 , d2 ) ., , X := X1 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 U2 : U1 X1 (x1 ), U2
X2 (x2 )}. CB4.3.2. . X X1 X2 X1 X2 X1 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).yx
, ,lim inf f (y) := lim f (y) := sup inf f (U )
yx
yx
U (x)
60
. 4.
f x ( (x)).C : x 6 dom f , (x, t) 6 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 . lim inf f (y) + f (x) lim inf f (y).
yx
yx
: (x, t) 6 epi f , t < lim inf f (y) = sup inf f (U ).yx
U (x)
, inf f (U ) > t U x. , (X R) \ epi f . B4.3.6. . , 4.3.5,
.4.3.7. f : X R , f f . CB4.3.8. f : X R , t R {f t}.C : x 6 {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. . B
4.4.
61
4.4. 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. . . CB4.4.4. . 4.4.3 , . . ,
.4.4.5. . .C . B4.4.6. (. . ).C , f : X R X . t :=0
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 . B4.4.7. . , (. 9.4.4).
62
. 4. 4.4.8. .C . B
4.4.9. . . CB4.5. 4.5.1. B X. {B 2 : B B} B X 2 .C (B1 B1 ) (B2
B2 ) (B1 B2 ) (B1 B2 ) B4.5.2. . F X UX X. F , F 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) .C (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). B4.5.5. . , ( ) 4.5.4 (1)4.5.4 (3), ( ) .4.5.6.
. , , , .C : 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 0 V d(x, y 0
) /2 , , d(x, y) d(x, y 0 ) + d(y 0 , y) , . ., , V B (x) , ,B (x)
F . B4.5.7. , B1 (x1 ) . . . Bn (xn ) Bn+1 (xn+1 ) . . . , (n ) , .
CB4.5.8. .C 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 =f (u1 ) f (u2 )
=(u1 ,u2 )U 2
= f U 2 f 1 f (f 1 V f ) f 1 = (f f 1 ) V (f f 1 ) V,, 1.1.6, f
f 1 = Iim f IY . B
64
. 4. 4.5.9. .C 4.5.8 4.5.4. B
4.5.10. X0 X (. . cl X0 = X) f0 : X0 Y X0 Y . , , f : X Y , f0 ,
. . , f |X0 = f0 .C 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 . Bb
db) 4.5.11. . f : (X, d) (X,b X X ( ), b (,d = db f . f X Xbb), f X
X , , im f = X.4.5.12. . (X, d) . b db) : (X, d) (X,b db) (X,bbbb
(X, d ). (X, d ) , (X, d)
- (X,b db)@
1@R@?b1 , db1 )(X
b1 , db1 ) X 1 : (X, d) (Xb1 , db1 ), (Xb db) (Xb1 , db1 ) :
(X,b Xb1 . XC 4.5.10. , 0 := 1 1 . 0 b 1 (X) Xb1 . (X) X
4.6.
65
b 0 X.b1 . xb1 . , Xb1 X (1 (xn )), xn X., (xn ) . , b xb ((xn
)) X.b := lim (xn ), xb X.1 (bx) = lim 0 ((xn )) = lim 1 ((xn )) =
lim 1 (xn ) = xb1 .b X. X X. X :bb := X / d((xx1 x2 d(x1 (n), x2
(n)) 0. X1 ), (x2 )) :=b .lim d(x1 (n), x2 (n)), : X Xb db) : (x)
:= (n 7 x (n : (X, d) (X,N)). Bb db), 4.5.13. . (X, 4.5.12, , (X,
d).4.5.14. . X0 (X, d) , (X0 , d|X02 ) (X, d).4.5.15. . . CB4.5.16.
X0 X. X0 X0 X.b := cl X0 : X0 Xb .C Xb, . X 4.5.15. 4.5.12. B4.6.
4.6.1. . CB4.6.2. . 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 .C 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 . B4.6.5. U X , V UX U1 , . . . , Un U , U = U1 . .
. Un U1 , . . . , Un V . CB4.6.6. . , 4.6.5, : , .4.6.7. . , .
CB4.6.8. C(X, F) X F 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 1 . Ud = fil {U : UF }. CB4.6.9. C(X,
F) . CB4.6.10. . E C(X, F) , E {g(X) : g E } F.
4.6.
67
C : , {g(X) : g E } , . E UF 0 0 0 0 . U0 - E 0 E . U UX , \U
:=f 1 0 ff E 0
(. 4.2.9). g E f E 0 , g f 1 0 ,0 = 01 (g f 1 )1 = (f 1 )1 g 1 =
f g 1 . , 4.6.8 g (U ) = g U g 1 g (f 1 0 f ) g 1 (g f 1 ) 0 (f g 1
) 0 0 0 . g , E .: 4.5.15, 4.6.7, 4.6.8 4.6.9 UF U - E . 0 UF , 0 0
0 , U UX , \
U
g 1 0 g
gE
( U E )., {U (x) : x X} X. X, {U (x0 ) : x0 X0 }. , 1.1.10IX
[
U (x0 ) U (x0 ) =
x0 X0
=
[(x0 ,x0 )IX0
U 1 (x0 ) U (x0 ) = U IX0 U.
68
. 4.
{g|X0 : g E } FX0 . , 0 -. , E 0 E , , g E f E 0 g IX0 f 1 0 . ,
g f 1 = g IX f 1 g (U IX0 U ) f 1 g (g 1 0 g) IX0 (f 1 0 f ) f 1 ==
(g g 1 ) 0 (g IX0 f 1 ) 0 (f f 1 ) == Iim g 0 (g IX0 f 1 ) 0 Iim f
0 0 0 . , 4.6.8, E 0 U - E . B4.6.11. . -. : , U , 0 /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) .C (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 . B4.7.4. .
4.7.3 (4) , () . .4.7.5. . X (f : X R) , sup{f (x) : } < + x X.
G X G0 , (f ) , . . supxG0 sup {f (x) : } +. CB4.7.6. . .C 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 6= nN Un , , x 6
G. ,x B0 (x0 ) G. . B4.7.7. . . . f : [0, 1] R x [0, 1) D+ f (x) :=
lim inf
f (x + h) f (x);h
D+ f (x) := lim sup
f (x + h) f (x).h
h0
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). . :
Xhh4n xii4nn=0
4.8.
71
( hhxii := (x [x]) (1 + [x] x) x ),
+X1sin (n2 x)2nn=0
, ,
X
bn cos (an x)
n=0
( a , 0 < b < 1 ab > 1 +
32 ).
4.8. 4.8.1. . , , R2 . , , , .4.8.2. . (= ) () . () . . .4.8.3.
. R2 . G1 G2 , G1 G2 = R2 \ ; = G1 = G2 . CB4.8.4. . G1 G2 , 4.8.3,
. , , . . . : .
72
. 4.
4.8.5. . D, D1 , . . . , Dn (= ) , Dm Dk = m 6= k D1 , . . . ,
Dn int D. n[D\int Dkk=1
. , (= ) , . .4.8.6. . F F . F () R2 F () () F . , 4.8.3 , ,
.4.8.7. K G , K. F , K int F F G. CB4.8.8. . F , 4.8.7, (K, G).4.1.
. , .4.2. X 2 , X?4.3. S [0, 1]
Z1d(f, g) :=
|f (t) g(t)|dt1 + |f (t) g(t)|
(f, g S)
0
( ?). .
73
4.4. , NN d(, ) = 1/ min {k N : k 6= k }.
, d NN .4.5. ? ?4.6. ? ?4.7. , .4.8. A B RN
d(A, B) :=
sup inf |x y|xA yB
sup inf |x y|
.
yB xA
, d . . ?4.9. , RN . ?4.10. , RN .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. X F p : X R . (1) dom p X;(2) p(x) 0 x X;(3) ker p
:= {p = 0} X;
(4) B p := {p < 1} Bp := {p 1} , p B
, B p B Bp ;
(5) X = dom p , B p .C x1 , x2 dom p 1 , 2 F, 3.7.6 p(1 x1 + 2
x2 ) |1 |p(x1 ) + |2 |p(x2 ) < + + (+) = +., (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). . B
5.1.
75
5.1.2. p, q : X R . p q( (R )X ) , Bp Bq .C : , {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.
B5.1.3. X, Y , T X Y p : Y R . , ,pT (x) := inf p T (x) x X. pT : X
R , BT := T 1 (Bp ) , pT = pBT .C x1 , x2 X 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 . , ,
x B pT , pT (x) = inf{p(y) : (x, y) T } < 1. , y
T (x) , p(y) < 1. , x T 1 (B p ) T 1 (Bp ) = BT .
, B pT BT BpT . 5.1.1 (4), : pBT = pT . B5.1.4. . 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 ), X, .5.1.7. M (R )X , (X, p) p M. CB5.1.8. .
MX ( ), x X, x 6= 0, p MX , p(x) 6= 0. X ( ) .5.1.9. . , , . X ( )
X kk () kkX , k| Xk, X. (X, k k) . , X.5.1.10. .(1) (X, p) (X,
{p}). .(2) M ( ) X. M , X.(3) (Y, N) T X Y , 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 . , p1 , 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)| : , xRN
() RN ( , . 10.11.6).(7) (X, kk) (Y, kk) ( F). T L (X, Y ) , . .
kT k := sup {kT xk : x X, kxk 1} = supxX
kT xk.kxk
78
. 5.
( , 0/0 := 0.), k k : L (X, Y ) R . , BX := {k kX 1}, T1 , T2 L
(X, Y ) 1 , 2 Fk1 T1 + 2 T2 k = sup k k1 T1 +2 T2 (BX ) == sup k
k((1 T1 + 2 T2 )(BX )) sup k1 T1 (BX ) + 2 T2 (BX )k |1 | sup k kT1
(BX ) + |2 | sup k kT2 (BX ) == |1 | kT1 k + |2 | kT2 k. B(X, Y ),
, , . , B(X, Y ) ( ). , T L (X, Y ) , , . . K, kT xkY K kxkX (x X).
kT k K, . CB(8) X F k k X. , , X 0 := B(X, F) , . . f :kf k =
sup{|f (x)| : kxk 1} = supxX
|f (x)|.kxk
X 00 := (X 0 )0 := B(X 0 , F) X . x X f X 0x00 := (x) : f 7 f
(x)., (x) (X 0 )# = L (X 0 , F). ,kx00 k = k(x)k = sup {|(x)(f )| :
kf kX 0 1} =
5.2.
79
= sup{|f (x)| : |f (x)| kxkX (x X)} == sup{|f (x)| : f ||(k kX
)} = kxkX . , , 3.6.5 3.7.9. , (x) X 00 x X. , : X X 00 , : x 7
(x), , kxk = kxk x X. X , , . , , x x00 := x , . . X X 00 . X , X X
00 ( ). . , , . , , C([0, 1], F). CB5.1.11. . , 5.1.10 (8), ( ) X X
0 . x X f X 0 ( f x) (x, f ) :=hx | f i := f (x). X 0 x0 , . . hx |
x0 i = (x, x0 ) = x0 (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
. CB5.2.2. . Up := fil {{dp t} : t R+ \ 0} (X, p).
80
. 5.
5.2.3. Up . (1) Up fil {IX };(2) U Up U 1 Up ;(3) ( U Up ) ( V
Up ) V V U . CB5.2.4. . (X, M) . U := sup{Up : p M} X ( UM , UX .
.). ( 5.2.3 (1) 1.3.13.)5.2.5. (X, M) U . (1) U fil {IX };(2) U U U
1 U ;(3) ( U U ) ( V U ) V V U .C (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 . B5.2.6. M X , UM , . . {V : V UM } = IX .C : (x, y) 6 IX , . .
x 6= y. p M p(xy) > 0. , (x, y) 6 {dp 1/2 p(xy)}. 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. B
5.2.
81
5.2.7. X UX (x) := {U (x) : U UX } (x X). (x) x X. (1) (x) fil
{x};(2) ( U (x)) ( V (x) & V U ) ( y V ) V (y).C (. 4.1.8).
B5.2.8. . : x 7 (x) (X, M), (x) x. :X , M , (UM ) . .5.2.9. x X X
(x) = sup{p (x) : p MX }. CB5.2.10. X . x X U (x) U x X (0).C 5.2.9
1.3.13 (X, p). > 0 {dp }(x) = Bp + x, Bp := {p 1}. , p(yx) , y =
(1 (yx))+x 1 (yx) Bp . , y Bp + x, p(y x) = inf{t > 0 : y x tBp
} . B5.2.11. . 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.
C : x1 6= 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. B5.2.13. . , , , ,,
, . .5.2.14. (X, p) X0 X. - (X/X0 , pX/X0 ) , X0 .C : x 6 X0 . (x)
6= 0, , , : X X/X0 . 0 6= :=pX/X0 ((x)) = p1 ((x)) = inf{p(x + x0 )
: x0 X0 }. , x + /2 Bp X0 , . . x X0 . ,X0 .: x - X/X0 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. . B5.2.15. - -.C U ( ) U 6= ( ). 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 . B5.3. 5.3.1. . M N . , M N, M N, UM UN . M N N 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 ).C (1) (2) (3) (4): .(4) (5): (.5.1.2), q pBp1 /1 . . . pBpn
/n =
1111=p1 . . . pn ... p1 . . . pn .1n1n(5) (1): , M {q} q N. V Uq
, V {dq } > 0. nnoo . . . dpn {dq } dp1 tt p1 , . . . , pn M t
> 0. , , Up1 . . . Upn =U{p1 ,...,pn } UM . , V UM . B5.3.3. .
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 .C 5.3.2 5.1.2. B5.3.5. . p, q : FN R FN . p q ker p
ker q. CB5.3.6. . . CB
84
. 5.
5.3.7. (X, M) (Y, N) T L (X, Y ) . :(1) N T M;(2) T (UX ) UY , T
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 .
CB5.3.8. (X, kkX ) (Y, kkY ) T L (X, Y ) . :(1) T (. . T B(X, Y
));(2) k kX k kY T ;(3) T ;(4) T ;(5) T .C 5.3.7. B5.3.9. . 5.3.7 ,
M - ( ) . M := {sup M0 : M0 M}. .5.3.10. . X := F X0 X0 := F1, 1 :
7 1 ( ). M := {p : }, p (x) := |x()| (x F ). , M X. : X X/X0 . , M1
. M1 .5.3.11. . (X, M) X0 X. M1 , : X X/X0 , MX/X0 . (X/X0 , MX/X0
)
5.4.
85
- X X0 .5.3.12. - X/X0 , X0 . CB5.4. 5.4.1. . (X, M) . (X, M) ,
d X, UM = Ud . X , M, X . X , , X .5.4.2. . , .C : 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 ) :=
X1 pk (x1 x2 )2k 1 + pk (x1 x2 )
k=1
( P kk=1 1/2 , d )., d . . , (t) := t(1 + t)1 (t R+ ). , 0 (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) =
XX11(p(xy))(pk (x z) + pk (z y)) kk22kk=1
k=1
X1((pk (x z) + (pk (z y))) = d(x, z) + d(z, y).2kk=1
Ud UM . , Ud UM . {d }, (x, y) {dp1 t} . . . {dpn t}. d(x, y)
=
nXX1 pk (x y)1 pk (x y)+k2 1 + pk (x y)2k 1 + pk (x y)
k=1
k=n+1
nX1t1t X 1++.1+t2k2k1 + t 2nk=1
1
k=n+1
n
t(1 + t) + 2 , n t 0, t n (x, y) {d }. , {d } UM , . , UM Ud .
pn M t > 0 > 0 , {dpn t} {d }., :=
1 t,2n 1 + t
1 t1 pn (x y) d(x, y) = nn2 1 + pn (x y)2 1+t x, y , pn (x y) t.
B
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 .C (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 6 nU . , 1/n xn 6 U
n N, . . (1/n xn ) .(3) (1): p M. n N, V nBp ., sup p(V ) sup p(nBp
) = n < +. B5.4.5. . , .C : .: 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 .
B5.4.6. . 5.4.5 , .5.5. 5.5.1. . .
88
. 5.
5.5.2. . . , , . CB5.5.3. , (= ) .PC : n=1 kxn k < + (xn ).
sn :=x1 + . . . + xn , m > k
m
m
X
X
xn kxn k 0.ksm sk k =
n=k+1
n=k+1
: (xn ) . (nk )kN , kxn xm k 2k n, m nk . xn1 + (xn2 xn1 ) +
(xn3 xn2 ) + . . . x,. . xnk x. , xn x. B5.5.4. X X0 X. - X/X0 .C :
X X := X/X0 . , x X1 P x (x) , 2kxk kxk kxk. , n=1 xn , PX , xn 1
(xn ), P n=1 kxn k. 5.5.3 x := n=1 xn . x := (x).
nnXX
x xk x xk
0.k=1
k=1
5.5.3, , X . B5.5.5. . , 5.5.3 . , (X, p) , - X/ ker p. CB
5.5.
89
5.5.6. . X, Y X 6= 0. B(X, Y ) , Y .C : (Tn ) B(X, Y ). x X kTm
xTk xk kTm Tk k kxk 0, . . (Tn x) Y . , T x := lim Tn x. , T . |kTm
k kTk k| kTm Tk k (kTn k) R, , . . supn kTn k < +. , kTn xk supn
kTn k kxk, : kT k < +. , kTn T k 0. > 0 n0 , kTm Tn k /2 m, n
n0 . , x BX m n0 , kTm x T xk /2. kTn x T xk kTn x Tm xk +kTm x T
xk kTn Tm k + kTm x T xk n n0 . ,kTn T k = sup{kTn x T xk : x BX }
n.: (yn ) Y . x X kxk = 1. 3.5.6 3.5.2(1), x0 ||(k k), (x, x0 ) =
kxk = 1., Tn := x0 yn : x 7 (x, x0 )yn B(X, Y ), kTn k = kx0 k kyn
k. , kTm Tk k = kx0 (ym yk )k= kx0 k kym yk k = kym yk k, . . (Tn )
B(X, Y ). T := lim Tn . kT x Tn xk = kT x yn k kT Tn k kxk 0. , T x
(yn ) Y . B5.5.7. . ( ) . CB5.5.8. . X , : X X 00 , X X 00 . cl (X)
X.C 5.5.7, X 00 . 5.1.10 (8) X X 00 . 4.5.16. B5.5.9. .(1) : ,
90
. 5.
, , 5.5.45.5.8.(2) E . x F E kxk := sup |x(E )|. l (E ) := l (E
, F) := dom k k 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 ).R(4) S :=
(E , X, ) . , X RE , RE X R , : X R (), . .RR# X+ xn 0, xn X xn (e)
0 e E . ,, f F E ( S) (, , ).RR Np (f ) := ( |f |p )1/p p 1, R ( ).
dom N1 . f F E Re f, Im f RE . , N1 (f ) = N (f ), N (g) :=
5.5. 91
Z:= inf sup 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 ).
11N1 (f g) Np (f )Np0 (g)= 1, p > 1 .+p p0C :0
xpypxy 0pp
(x, y R+ ),
|f |/Np (f ) |g|/Np0 (g) , Np (f ) Np0 (g) . Np (f )Np0 (g) = 0
. B Lp := dom Np .C |f +g|p (|f |+|g|)p 2p (|f ||g|)p = 2p (|f |p
|g|p ) 2p (|f |p +|g|p ) B Np , Np (f + g) Np (f ) + Np (g).C p = 1
. p > 1 Np (f ) = sup{N1 (f g)/Np0 (g) : 0 < Np0 (g) < +}
(f Lp ), . 0 , Np (f ) > 0 g := |f |p/p g Lp0 , , Np (f ) = N1
(f g)/Np0 (g).R0 , N1 (f g) = |f |p/p +1 = Np (f )p , p/p0 + 1
=RR00p (1 1/p) + 1 = p. , Np0 (g)p = |g|p = |f |p =0Np (f )p , Np0
(g) = Np (f )p/p . 0
N1 (f g)/Np0 (g) = Np (f )p /Np (f )p/p =0
0
= Np (f )pp/p = Np (f )p(11/p ) = Np (f ),
92
. 5.
. B dom N1 . f F E Re f, Im f RE . , , N (g) :=
Z:= inf sup xn : (xn ) X, xn xn+1 , ( e E ) |g(e)| lim xn
(e)n
n
g F E . F := R, dom N1 , , X (dom N, N ).- Lp / ker Np , - kkp ,
, () p- , p- Lp . ,R Lp (S), Lp (E , X, ) . ., S (, A , ), Lp (, A
, ), Lp (, ) Lp (), . . Lp .C . - PPn P t := k=1 Np (fk ), fk Lp .
n := k=1 fkn sn :=k=1 |fk |. , (sn ) . R (spn ). , spn tp < +. e
E g(e) := lim spn (e) , g L1 . h(e) := g 1/p (e), , h Lp sn (e)
h(e) e E . |n | P sn h , e E k=1 fk (e). f0 (e) |f0 (e)| h(e), , ,
, f0 Lp . (= ), :
5.5.
93
1/pR|n f0 |p 0. , Np (n f0 ) = (Lp , Np ) . 5.5.35.5.5. B PS E ,
. . R ,X:=x :=eE R R Px(e),L,peEp- . lp (E ).
Pp 1/p. E := N lp kxkp :=eE |x(e)| , 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 7 1 (q Q) ( ). RE 1 l (E ).R S := E , X, 1 E F, N (f ) := inf{t
> 0 : |f | t1} < +, . L .
94
. 5.
- L / ker N L , k k . L , , ( L ) . L . CB L , , C(Q, F), lp (E
),c0 (E ), c, lp , Lp (p 1), . , . . , L1 ( - ). , X , X 0 Lp p
1.R(6) S := E , X, p 1. , e E Q (Ye , k kYe ). f eE Ye |||f ||| : e
7 kf (e)kYe . , , Np (f ) := inf{Np (g) : g Lp , g |||f |||}. , dom
Np Np .- dom Np / ker Np ||||||p (Ye )eE p (, Lp S). p .PC k=1PnNp
(fk ) < +. (sn := k=1 |||fk |||) g Np (g) < +. , e E (sPn
(e)), . .P k=1 kfk (e)kYe . - Ye , k=1 fk (e) f0 (e) Ye e E . kf0
(e)kYe g(e) Pn e PE, , f0 dom Np . , Np ( k=1 fk f0 ) k=n+1 Np (fk
) 0. B E := N Y (Yn )nN Y := (Y1 Y2 . . . )p , p . y Y -
5.5.
95
(yn )nN , yn Yn
|||y|||p :=
X
!1/pkyn kpYn
< +.
k=1
, Ye := X e E , X F, Fp := dom Np Fp :=Fp / ker Np . X- E ( , p-
). , Fp . , Fp (, Fp 6= Lp ). Fp , . , Fp , , , . Fp Lp ( :Lp (X),
Lp (S, X), Lp (, A , ), Lp (, ) . .) X- , p- , p- X- . , Lp (X) . p
= 1. , f :Xf=f 1 (x) x,ximf
f 1 (x) x im f . ,Z
Z|||f ||| =
X
kf 1 (x) xk =
ximf
Z=
Xximf
f 1 (x) kxk =
Xximf
Zkxk
f 1 (x) < +.
96
. 5.
f X :ZX Zf :=f 1 (x) x.ximf
R , , , . , ,
Z X ZX Z
f = 1 (x) x f 1 (x) kxk =f
ximf
ximf
Z=
X
Zkxkf 1 (x) =
|||f |||.
ximf
R 4.5.10 5.3.8 B(LRR 1 (X), X). ( E . .) .(7) , . , . . (xn ) .
( ) (kxn k), . . (xn ). (xn ) P (xn ) () . + . . . ). n=1 kxn k
< + (= x1 + x2 P 5.5.3 x=n=1 xn : x = lim s ,P s :=n xn ( ) , N.
x (xn ), (xn ) x (P: x = nN xn ). : ( ). dim X < + (= ). .
5.6.
97
. X P 2 (tn ) , n=1 tn < +, (xn ), kxn k = tn n N. (X, M) . ,
(xe )eE P ( x) x := eE xe , x (X, M) (s ), E , P. . s x (X, M). p
eE p(xe ), , (xe )eE (, , , ). Y T B(X, Y). T L1 (X) L1 (Y), X- f ,
T f : Re 7 T f (e) R e E . f L1 (X) T f L1 (Y) E T f = T E f . : .
CB5.6. 5.6.1. X, Y , Z , T L (X, Y ) S L (Y, Z) . kST k kSk kT k, .
. .C x X kST xk kSk kT xk kSk kT k kxk. B5.6.2. . , , () F. A F, ()
:(a, b) 7 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 6= 0 B(X) IX kIX k = 1. B(X) , X .C X = 0, . X
6= 0, 5.5.6. B5.6.5. . 5.6.4 IX , F, . ( , 1 = I0 = 0!) X 6= 0 F
FIX .5.6.6. . X T B(X). r(T ) := inf kT n k1/n : n N T . ( (.
8.1.12).)5.6.7. r(T ) kT k.C , 5.6.1, kT n k kT kn . B5.6.8. r(T )
= lim
pn
kT n k.
C > 0 s N , kT s k (r(T ) + )s . n N n s n = k(n)s+l(n),
k(n), l(n) N 0 l(n) s 1. ,kT n k = kT k(n)s T l(n) k kT s kk(n) kT
l(n) k
5.6.
1 kT k . . . kT s1 k kT s kk(n) = M kT s kk(n) .
99
,r(T ) kT n k1/n M 1/n kT s kk(n)/n M 1/n (r(T ) + )k(n)s/n = M
1/n (r(T ) + )(nl(n))/n . M 1/n 1 (n l(n))/n 1, r(T ) lim supkT n
k1/n r(T ) + . lim inf kT n k1/n r(T ) . . B5.6.9. . X T B(X). :(1)
1 + T + T 2 + . . . B(X);(2) kT k k < 1 k N;(3) r(T ) < 1.Pk
(1)(3) =k=0 T1(1 T ) .C (1) (2): , (T k ) .(2) (3): .(3) (1): 5.6.8
> 0 k 1/k k N r(T ) + < 1.P r(T k) kT k , kTkk=0.B(X)5.5.3,,P
k=0 T k P B(X).Pn 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 . B
100
. 5.
5.6.10. . kT k < 1, (1 T ) () (= ), . . . k(1 T )1 k (1 kT
k)1 .C , k(1 T )1 k
X
kT k k
k=0
X
kT kk = (1 kT k)1 . B
k=0
5.6.11. . k1 T k < 1, T k1 T 1 k
k1 T k.1 k1 T k
C 5.6.9,1+
X
(1 T )k =
X
(1 T )k = (1 (1 T ))1 = T 1 .
k=0
k=1
:
X
XX
kkk1 T kk . Bk(1T)k(1T)kT 1 1k =
k=1
k=1
k=1
5.6.12. . X Y . () . T 7 T 1 .C S, T B(X, Y ) , T 1 B(Y, X), ,
kT 1 k kS T k 1/2. T 1 S B(X). k1 T 1 Sk = kT 1 T T 1 Sk kT 1 k kT
Sk
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.
101
,kS 1 k kT 1 k kS 1 T 1 k == kS 1 (T S)T 1 k kS 1 k kT Sk kT 1
k
1 1kS k.2
kS 1 k 2kT 1 k. kS 1 T 1 k kS 1 k kT Sk kT 1 k 2kT 1 k2 kT Sk.
B5.6.13. . X F T B(X). F T , ( T )1 B(X). R(T, ) := ( T )1 R(T, ) (
T ). res(T ). 7 R(T, ) res(T ) B(X) T . F \ res(T ) T Sp(T ) (T ).
.5.6.14. . X = 0, T = 0 B(X) . , X 6= 0. X 6= 0 F := R , F := C (.
8.1.11). CB5.6.15. res(T ) , 0 res(T ), 0 R(T, ) =
X
(1)k ( 0 )k R(T, 0 )k+1 .
k=0
|| > kT k, res(T )
R(T, ) =
1 X Tk,kk=0
kR(T, )k 0 || +.
102
. 5.
C k( T ) (0 T )k = | 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 ==
X
(1)k ( 0 )k R(T, 0 )k+1 .
k=01
5.6.9 || > kT k (1 T /) , . .1R(T, ) =
T1
1
,
1 X Tk=.kk=0
kR(T, )k
11. B|| 1 kT k/||
5.6.16. T .5.6.17. . , || >r(T ) P R(T, )= k=0 T k /k+1 , (.
8.1.12).5.6.18. S T , S T .C : ST = T S S( T ) = S ST = S T S = ( T
)S R(T, )S( T ) = S R(T, )S = S R(T, ) ( res(T )).: SR(T, 0 ) =
R(T, 0 )S S = R(T, 0 )S(0 T ) (0 T )S = S(0 T ) T S = ST . B
5.6.
103
5.6.19. , res(T ), (= )R(T, ) R(T, ) = ( )R(T, )R(T, ).C = ( T )
( T ) R(T, ) , R(T, ) , . B5.6.20. , res(T ), R(T, )R(T, ) = R(T, )
R(T, ). CB5.6.21. res(T ) dkR(T, ) = (1)k k! R(T, )k+1 .
CBdk5.6.22. . Sp(ST ) Sp(T S) .C , 1 6 Sp(ST ) 1 6 Sp(T S). , 6
Sp(ST ) 6= 0 1 6
1Sp(ST ) 1 6 Sp
1ST
1 6 Sp
1TS
6 Sp(T S).
, 1 6 Sp(ST ). (1 ST )1 1 + ST + (ST )(ST ) + (ST )(ST )(ST ) +
. . . ,T (1 ST )1 S T S + T ST S + T ST ST S + . . . (1 T S)1 1 ,
(1 T S)1 = 1 + T (1 ST )1 S
104
. 5.
( 1 6 Sp(T S)). :(1 + T (1 ST )1 S)(1 T S) == 1 + T (1 ST )1 S T
S + T (1 ST )1 (ST )S == 1 + T (1 ST )1 S T S + T (1 ST )1 (1 ST
1)S == 1 + T (1 ST )1 S T S + T S T (1 ST )1 S = 1;(1 T S)(1 + T (1
ST )1 S) == 1 T S + T (1 ST )1 S + T (ST )(1 ST )1 S == 1 T S + T
(1 ST )1 S + T (1 ST 1)(1 ST )1 S == 1 T S + T (1 ST )1 S + T S T
(1 ST )1 S = 1 , . B5.1. , , .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)0 = Lp0 (X 0 ), X .5.18. (Xn
)
(X0 :=
x
)Y
Xn : kxn kXn 0
nN
c0 ( kxk := sup{ kxn k : 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 7 f (x, y) H # y
Y ;(2) f (x, y) = f (y, x) x, y H, 7 F, . . .6.1.2. . , f x H f (x,
) : y 7 (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)
C
(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))
B
6.1.
107
6.1.4. . f , , f (x, x) 0 x H. : (x, y) := hx | yi := 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).C (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) 6= 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) 6= 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). B6.1.6. ( , ) H, k k : x 7 (x, x)1/2 H.C .
, kx + yk2 = (x, x) + (y, y) + 2 Re(x, y) (x, x) + (y, y) + 2kxk
kyk = (kxk + kyk)2 . B6.1.7. . H ( , ) kk . H , (H, k k) .
108
. 6.
6.1.8. H kx + yk2 + kx yk2 = 2(kxk2 + kyk2 ) (x, y H) .C kx +
yk2 = (x + y, x + y) = kxk2 + 2 Re(x, y) + kyk2 ;kx yk2 = (x y, x
y) = kxk2 2 Re(x, y) + kyk2 B6.1.9. . (H, k k) , H , . . , , ( , )
H , kxk = (x, x)1/2 x H.C HR H x, y HR (x, y)R :=
1kx + yk2 kx yk2 .4
, ( , y)R (x1 , y)R + (x2 , y)R =
1kx1 + yk2 kx1 yk2 + kx2 + yk2 kx2 yk2 =4
1=kx1 + yk2 + kx2 + yk2 kx1 yk2 + kx2 yk2 =4
1 1=(k(x1 + y) + (x2 + y)k2 + kx1 x2 k2 4 2=
1k(x1 y) + (x2 y)k2 + kx1 x2 k2 ) =2
1 11=kx1 + x2 + 2yk2 kx1 + x2 2yk2 =4 22
6.1.
122=k(x1 + x2 )/2 + yk k(x1 x2 )/2 yk =2= 2 ((x1 + x2 )/2, y)R
.
109
, 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) := Re1 (( , y)R )(x), Re1 (. 3.7.5). F := R ,
(x, y) = (x, y)R = (y, x) (x, x) =kxk2 , . . . 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) ,
1kx + iyk2 kx iyk2 =4
1=|i| ky ixk2 |i| kix + yk2 = (ix, y)R .4 ,(x, iy)R =
(x, x) = (x, x)R i(ix, x)R =
ikix + xk2 kix xk2 == kx2 k 4
i222= kxk 1 |1 + i| |1 i|= kxk2 .4 6.1.3. B
110
. 6.
6.1.10. .(1) L2 ( - R). : (f, Pg) := f g f, g L2 . , l2 (E ) (x,
y) := eE xe ye x, y l2 (E ).(2) H ( , ) :H 2 F H. , HR ( , )R : (x,
y) 7 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 k k k
k 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, k k) k k. , H/H0
, , H. H/H0 , (, ). .(5) (He )eE H 2, . . h H
6.2.
111
, h := (he )eE , he He e E , !1/2X2< +.khk :=khe keE
5.5.9 (6), H . f, g H, ,
1kf + gk2 + kf gk2 =21=2
X
2
kfe + ge k +
eE
=
X
2
kfe ge k
=
eE
X 1
kfe + ge k2 + kfe ge k2 =2
eE
=
X
kfe k2 + kge k2 = kf k2 + kgk2 ,
eE
, , H . H (He )eE eE He . E := N H := H1 H2 . . . .(6) H S . L2
(S, H), H- , , . CB6.2. 6.2.1. U (r + )BH \ rBH , 0 < r, H. :
diam U 12r.C x, y U , , 1/2(x + y) U , ,
2kx yk2 = 2 kxk2 + kyk2 4 k(x + y)/2k
112
. 6. 4(r + )2 4r2 = 8r + 42 12r. B
6.2.2. . U H x H \ U . , , u0 U, kx u0 k = inf{kx uk : u U }.C U
:= {u U : kx uk inf kU xk + }. 6.2.1 (U )>0 U . B6.2.3. . 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 .C (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 0 (x0 ) = 2(x0 x,
). B6.2.5. . 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 .C 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 ) H0 + H0 . B
6.2.
113
6.2.7. . H0 H0 H0 PH0 .6.2.8. . x y kx + yk2 = kxk2 + kyk2 .
CB6.2.9. . : H 6= 0, H0 6= 0 kPH0 k = 1. CB6.2.10. . P L (H) , P 2
= P , :(1) P H0 := im P ;(2) khk 1 kP hk 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.C (1) (2):
6.2.9.(2) (3): H1 := ker P = im P d . x H1 . x = P x + P d x x P d
x, kxk2 kP xk2 = (x P d x, x P d x) = (x, x) 2 Re(x, P d x) + (P d
x, P d x) = kxk2 + kP d xk2 . P d x = 0, . . x im P . H1 = ker P H1
im P 6.2.6 : H1 = im P = H0 . , (P x, P d 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 h0 , h0 P h0 ) = 0, .
. h0 im P . x H h0 H0 (x P x, h0 ) = (x P x, P h0 ) = (P (x P x),
h0 ) = (P x P 2 x, h0 ) =(P x P x, h0 ) = 0. , 6.2.4, P x = PH0 x.
B6.2.11. P1 , P2 , P1 P2 = 0. P2 P1 = 0.C P1 P2 = 0 im P2 ker P1 im
P1 = (ker P1 ) (im P2 ) =ker P2 P2 P1 = 0 B
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 6= m.C : , P0 6.2.10 kP0 xk2
= (P0 x, P0 x) = (P02 x, x) =(P0 x, x). , x H l 6= m kPl xk2 + kPm
xk2
nX
kPk xk2 =
k=1
nX
(Pk x, x) = (P x, x) = kP xk2 kxk2 .
k=1
, x := Pl x, kPl xk2 + kPm Pl xk2 kPl xk2 kPm Pl k = 0.: , P .
,2
P =
nXk=1
!2Pk
=
nnXXl=1 m=1
Pl Pm =
nX
Pk2 = P.
k=1
, 6.2.10 (4), (Pk x, y) = (x, Pk y) , ,(P x, y) = (x, P y).
6.2.10 (4). B6.2.14. . 6.2.13 .6.3. 6.3.1. . (xe )eE H , e1 6=e2
xe1 xe2 . E H , (e)eE .
6.3.
115
6.3.2. . (xe )eE () , (kxe k2 )eE .
2
XX
kxe k2 .xe =
eEeEPC s := eE Pxe , E . 6.2.8, ks k2 = e kxe k2 . , 0 , , Xks0
s k2 = ks0 \ k2 =kxe k2 .e 0 \
, (s ) (kxe k2 )eE . 5.5.3, . B6.3.3. . (Pe )eE H. x H ()P (Pe
x)eE . P x := eE Pe x ()XX2H :=xe : xe He := im Pe ,kxe k < +
.eE
eE
PC E s := e Pe . P6.2.13, s . , 6.2.8,22ks xk2 =e kPe xk kxk x
H. 2, (kPe xk )eE ( ). P x :=PeE Pe x, . . P x = lim s x. P 2 x =
lim s P x = lim s lim0 s0 x = lim lim0 s s0 x =lim lim0 s0 x= lim s
x = P x. kP xk = k lim s xk =lim ks xk kxk , , P 2 = P . 6.2.10, ,
P im P .P xP im P , . . P x = x, x =eE Pe x eE kPe xk2 = kxk2 = kP
xk2 < +. Pe x
116
. 6. P2He (e E ),P x H . xe He eE kxe k < + , x := eE xe (PP)
x = eE xe = eE Pe xe = P x, . . x im P ., im P = H . B6.3.4. . H
(He )eE . , , , .6.3.5. . h H :khk = 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.
hhi. , hhi :x 7 (x, h)h (x H).6.3.6. . ( ),, -, , -, . .6.3.7. E H
x H (heix)eE () . :kxk2
X
|(x, e)|2 .
eE
C ,
2
2
X
X
X
2kxk heix = (x, e)e =k(x, e)ek . B
2
eE
eE
eE
6.3.
117
6.3.8. . E H ( H), P x H x = eE heix. , .6.3.9. E H , L (E ) H.
CB6.3.10. . , E , E = 0.6.3.11. . , .PPeE heih =eE (h, e)e =P C : h
E . h =0=0.eEP: x H, 6.3.3 6.2.4, x eE heix E . B6.3.12. . .C H E .
h H \ H0 , H0 := cl L (E ), h1 := h PH0 h E , , H 6= 0 E {kh1 k1 h1
} = E . . H = 0 . B6.3.13. . , H . H.6.3.14. . (xn )nN H. x0 := 0,
e0 := 0, yn := xn
n1Xk=0
hek ixn ,
en :=
ynkyn k
(n N).
118
. 6.
, (yn , ek ) = 0 0 k n 1 (, 6.2.13). , yn 6= 0, H. (en )nN , , ,
(xn )nN . , , , , . . . CB6.3.15. . E H x H. xb := (bxe )eE F E ,
xbe := (x, e), x ( E ).6.3.16. . E H. F : x 7 xb ( E ) H l2 (E ). F
1 : l2 (E ) HP1 F (x) := eE xe e x := (xe )eE l2 (E ). x, y H X(x,
y) =xbe ybe .eE
C l2 (E ). 6.3.3, b . , b 1 (x) = x . , F 1 xb = x x H F\x l2 (E
), . Xkxk2 =kbxe k2 = kbxk22 (x H)eE
. !XXXX(x, y) =xbe e,ybe e =xbe ybe0 (e, e0 ) =xbe ybe . BeE
eE
e,e0 E
eE
6.3.17. . , . ,
6.4.
119
, . . , . ( ).6.4. 6.4.1. . H . x H x0 := ( , x). x 7 x0 H H 0
.C , x = 0 x0 = 0. x 6= 0, kx0 kH 0 = sup |(y, x)| sup kyk kxk
kxk;kyk1
kyk1
kx0 kH 0 = sup |(y, x)| |(x/kxk, x)| = kxk.kyk1
, x 7 x0 H H 0 . , . l H 0 H0 := ker l 6= H ( l , ). kek = 1 , e
H0 , grad l := l(e) e. x H0 , (grad l)0 (x) = (x, grad l) = (x,
l(e) e) = l(e) (x, e) = 0., F x H 2.3.12 (grad l)0 (x) = l(x). , x
:= e (grad l)0 (e) = (e, grad l) = l(e)(e, e) = l(e),. . = 1.
B6.4.2. . , H 0 x 7 x0 H H 0 . l 7 grad l. 6.4.1 .
120
. 6.
6.4.3. .C : H H 00 , . . H H 00 , x00 (l) = (x)(l) = l(x), x H l
H 0 (. 5.1.10(8)). , . f H 00 . y 7 f (y 0 ) y H. , H , , x H = H ,
(y, x) = (x, y) = f (y 0 ) y H. (x)(y 0 ) = y 0 (x) = (x, y) = f (y
0 ) y H. y 7 y 0 H 0 , (x) = f. B6.4.4. H1 , H2 T B(H1 , H2 ). , ,
T : H2 H1 , x H1 , y H2(T x, y) = (x, T y). T B(H2 , H1 ) kT k = kT
k.C y H2 . x 7 (T x, y) y 0 T , . . H1 . x H1 , x0 = y 0 T . T y :=
x. , T L (H2 , H1 ). , , |(T y, T y)| = |(T T y, y)| kT T yk kyk kT
k kT yk kyk., kT yk kT k kyk y H2 , . . kT k kT k. T = T := (T ) ,
. . kT k = kT k kT k. B6.4.5. . 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) kT T k = kT k2 .
6.4.
121
C (1)(3) . kxk 1, kT xk2 = (T x, T x) = |(T x, T x)| = |(T T x,
x)| kT T xk kxk kT T k. , 6.4.4, kT T k kT k kT k = kT k2 , (4).
B6.4.7. H1 , H2 , H3 , T B(H1 , H2 ) S B(H2 , H3 ). (ST ) = T S .C
(ST x, z) = (T x, S z) = (x, T S z) (x H1 , z H3 ) B6.4.8. . TT H1
H2 . H1 H2 . , , , .6.4.9. . , .C 6.4.7 6.4.6 (1). B6.4.10. . T
B(H1 , H2 ) T B(H2 , H1 ). T , T . T 1 = T 1 . CB6.4.11. . T B(H)
Sp(T ) Sp(T ). CB6.4.12. (. 7.6.13). T
Tk+1
k. . . Hk1 Hk Hk+1 . . .
, T
Tk+1
k. . . Hk1 Hk Hk+1 . . . . CB
122
. 6.
6.4.13. . -( F) A , . . a 7 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 ,
ka ak = kak2 a A, C -.6.4.14. B(H) H C - ( ). CB6.5. 6.5.1. . H F T
B(H). T ( ), T = T .6.5.2. . T kT k = sup |(T x, x)|.kxk1
C t := sup{|(T x, x)| : kxk 1}. , |(T x, x)| kT xk kxk kT k, kxk
1. , t kT k. T = T , (T x, y) = (x, T y) = (T y, x) = (y, T x) ,. .
(x, y) 7 (T x, y) . , 6.1.3 6.1.84 Re(T x, y) = (T (x + y), x + y)
(T (x y), x y) t(kx + yk2 + kx yk2 ) = 2t(kxk2 + kyk2 ). T x = 0,
kT xk t. T x 6= 0. kxk 1 y := kT xk1 T x
TxTx,=kT xk = kT xkkT xk kT xk
1 2= (T x, y) = Re(T x, y) t kxk2 + kT x/kT xk k t,2. . kT k =
sup{kT xk : kxk 1} t. B
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)0 . , 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) ( , . 7.4.7).6.5.4. . T , inf kx T xk = 0.
kxk=1
C : t := inf{kxT xk : x H, kxk = 1} > 0. , / Sp(T ). x H kx T
xk tkxk. , -, ( T ) , -, H0 :=im(T ) ( k(T )xm (T )xk k tkxm xk k,
. . ) , , -, ( T )1 B(H), H = H0( kR(T, )k t1 ). , , H 6= H0 . y
H0
