. . .

. . C. C.

:

2006

Copyright & A K-C

517.98 94

..

: - . . -,

- ..

. ., .. : ..: , 2006.

560 . -

. : , - . - , .

1992 . . 1995 . - Kluwer Academic Publishers , .

, - .

c , 2006c ( ), 2006

c , 2006

c , 2006c .. , 2006c .. , 2006

Copyright & A K-C

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1. . . . . . . . . . . . . . . . . . 91.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531.6. . . . . . . . . . . . . . . . . . . . . 621.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

2. . . . . . . . . . . . . . . . . . . . . . . 802.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 802.2. . . . . . . . . . . . . . . . . 942.3. , . . . . . . . . 1062.4. . . . . . . . . . . . . . . . . . . . 1192.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1302.6. ,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1402.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

3. . . . . . . . . . . . . . . . . . . . . . . . . . . 1583.1. . . . . . . . . . . . . . . . . . . . . . . . 1593.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1703.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1833.4. . . . . . . . . . . . . . . . . . . . 1963.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2043.6. , . . . . . . . . . . . . 2133.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

4. . . . . . . . . . . 2294.1. . . . . . . . . . . . . . . . . . . . . . . . . . 2304.2. . . . . . . . . . . . . . . . . . . . . . . . . . 2424.3. . . . . . . . . . . . . 2534.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2634.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2724.6. . . . . . . . . . . . . . . . . . . . . 2834.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

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4

5. . . . . . . . . . . . . . . . . . . . . 3045.1. . . . . . . . . . . . . . . . . . . . . . . 3055.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3145.3. . . . . . . . 3215.4. . . . . . . . . . . . . . . . . 3285.5. . . . . . . . . . . . . . . . . . . . . . . 3335.6. . . . . . . . . . . . . . . . . . . . . . . . 3465.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3606.1. . . . . . . . . . . . . . . . . . . . . . . . . . . 3616.2. . . . . . . . . . . . . . . . . . . . . . 3696.3. , . . . . . . . 3766.4. . . . . . . . . . . . . . . . . . . . 3856.5. . . . . . . . . . . . . . . . . . . . . . . . . . 3946.6. . . . . . . . . . . . . . . . . . . . . . . . . . 4026.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410

7. . . . . . . . . . . . . . . . . 4187.1. . . . . . . . . . . . . . . . . . . . . . . 4197.2. . . . . . . . . . . . . . . 4267.3. . . . . . . . . . . . . . . . . . . . 4357.4. , . . . . . . . 4437.5. . . . . . . . . . . . . 4527.6. . . . . . . . . . . . . . . . . . . . 4597.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

1. . . . . . . . . . . . . . . . . . . . . . . . . . . 474

2. . . . . . . . . . . . . . . . . . . . 481

3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486

4. . . . . . . . . . . . . . . . . . . . . . . . 492

5. . . . . . . . . . . . . . . . . . . 498

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544

Copyright & A K-C

. - .

- .

x X, f(x) inf .

X , f : X R -, , , . , - inf f(X) , , x X, f(x) = inf f(X) ( ).

, . . , . -, , f . , - , . l, :

x X, f(x) l(x) inf .

, , , l X, . . X#. - .

, f , , . , - , f : X# R,

f(l) := supxX

(l(x) f(x)).

f f ., f(0) - .

Copyright & A K-C

6

, . . (f G), G : Y X , Y X.

, f l. - , - f . , , f x -

f(x) :={l X# : (x X) l(x) l(x) f(x) f(x)} ==

{l X# : l(x) = f(x) + f(l)},

. x , :

0 f(x).

, , f(x). , - (f G)(y). G X . , :

f1 + f2 = + (f1, f2);(f1, f2) : X R2, (f1, f2)(x) := (f1(x), f2(x)),

R2 . , ,

. , - , - . .

, . , - . - , . , . .

Copyright & A K-C

7

. . -, .

, . - . , . . , , .

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

, , . , - , . , - . - , .

1987 - . 1992 - , , 1995 Kluwer Academic Publishers, - . . . . . , 2002 2003 .

, - . . . -

Copyright & A K-C

8

-.

1986 , .

1999 , .

.

. .

Copyright & A K-C

1

- . , - . , , - .

- , . - -.

, - . -. . , . -.

, , -. - . , , , , - . - - .

, - - .

- K-, . . -, .

K- . , -

Copyright & A K-C

10 1.

, . . - , .

- , .

- , . - , - - , , , . .

R , N :={1, 2, . . . } .

1.1.

-, .

1.1.1. X . - R2. C X --, x y x+y, (, ) . - X P(X). , C P(X) , (, ) C+C C. C := {x : x C} C+D := {x+y : x C, y D}. -.

(1) - -.

. , E ,

A,B E C E , A C B C.(2)

- -. E P(X) . D :=

E .

x, y D (, ) . D A B E x A y B. C E , A C B C. , x y

Copyright & A K-C

1.1. 11

C E . C -, x + y C D, .

(3) X C X. C :=

C X :=

X. C P(X)

, C P(X) . x = (x), y = (y) C (, ) . , x + y C

, x + y C . .

(4) - -.

. (5) C D - , R,

C C +D -. (3) (4), C = (C) C+D = +(CD), (x) := x

+(x, y) := x+ y. 1.1.2. -, .

(1) := R2, - X () X.

(2) := {(, ) R2 : + = 1}. - - . X0 X x X, L := x + X0 := {x} + X0 - , X0. , L X0 := L x := L + (x), x L, L .

(3) := R+ R+, - , , . , K X , K +K K K K R+. R+ := {t R : t 0}.

(4) := {(, 0) R2 : || 1}. - . , C X, C C || 1.

(5) := {(, ) R2 : 0, 0, + = 1}. - - . , . , () .

(6) := {(, ) R2 : 0, 0, + 1}, - . - , .

(7) := {(, ) R2 : || + || 1}. - . .

Copyright & A K-C

12 1.

(8) := {(1, 0)}, - . , C , C = C. . , - , .

1.1.3. P(X) := P(X) - X. M P(X)

H(M) :={

C P(X) : C M}.

1.1.1 (1) H(M) -. - - M . , - M ( ) -, M . - H M H(M), M P(X). .

(1) H , . . A,B P(X) A B H(A) H(B).

. (2) H , . . H H = H.

, H(M) - M .

(3) P(X) - H, . .

C P(X) H(C) = C (M P(X))C = H(M).

-, .

(4) M P(X)

H(M) ={

H(M0) : M0 Pfin(M)},

Pfin(M) M . A , .

H(M) A (1). , A -, M A. H(M1)H(M2) H(M1 M2) , {H(M0) : M0 Pfin(M)} . 1.1.1 (2) A P(X).

(5) P(X), , (-) . - X , - - .

, - P(X) P(X) .

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1.1. 13

1.1.4. -, , - . , , -. , - - - . , 1.1.2. , , 1.1.4 (k) , 1.1.2 (k). M X x1, . . . , xn X.

(1) lin(M) := H(M) M . -

lin(M) =

{m

k=1

kxk : 1, . . . , m R, x1, . . . , xm M, m N}.

, :

lin({x1, . . . , xn}) ={

nk=1

kxk : 1, . . . , n R}.

lin() := {0}. . 1x1 + . . . + nxn - x1, . . . , xn. , M M .

(2) aff(M) := H(M) M . , x M , aff(M) x = lin(M x). , 0 M , aff(M) = lin(M).

aff(M) =

{m

k=1

kxk : 1, . . . , m R,m

k=1

k = 1, x1, . . . , xm M, m N}.

, :

aff({x1, . . . , xn}) ={

nk=1

kxk : 1, . . . , n R, 1 + . . .+ n = 1}.

aff({x, y}) x = y , x y. 1x1 + . . . + nxn x1, . . . , xn, 1 + . . .+ n = 1. , M M .

(3) cone(M) := H(M) M .,

aff(cone(M)) = lin(cone(M)) = cone(M) cone(M).

Copyright & A K-C

14 1.

, K , K K -, K. K -, K, , K (K).

cone(M) =

{m

k=1

kxk : 1, . . . , m R+, x1, . . . , xm M, m N}.

,

cone({x1, . . . , xn}) ={

nk=1

kxk : 1, . . . , n R+}.

{x} x = 0 , x ( , x). 1 0, . . . , n 0, 1x1+ . . .+nxn - x1, . . . , xn. , M M .

(4) bal(M) := H(M) - M . ,

bal(M) =

{M : || 1}.

(5) co(M) := H(M) M . - {x, y} x = y x y , x y. ( x = y .) M , , .

co(M) =

{m

k=1

kxk : 1, . . . , m R+,m

k=1

k = 1, x1, . . . , xm M, m N}.

,

co({x1, . . . , xn}) ={

nk=1

kxk : 1, . . . , n R+, 1 + . . .+ n = 1}.

1x1 + . . .+nxn x1, . . . , xn, 1 0, . . . , n 0 1 + . . . + n = 1. , - M M .

(6) sco(M) := H(M) . - sco co sco(M) = co(M {0}). ,

sco({x1, . . . , xn}) ={

nk=1

kxk : 1, . . . , n R+, 1 + . . .+ n 1}.

Copyright & A K-C

1.1. 15

(7) aco(M) := H(M) - M . aco = co bal. , , , .

aco(M) =

{m

k=1

kxk : 1, . . . , m R,m

k=1

|k| 1, x1, . . . , xm M, m N}.

,

aco({x1, . . . , xn}) ={

nk=1

kxk : 1, . . . , n R, |1|+ . . .+ |n| 1}.

1x1 + . . . + nxn x1, . . . , xn, |1|+ . . .+ |n| 1. , - M M .

(8) sim(M) := M (M) M . - simh := co sim. , simh = aco, . . M , M . C - sk(C), C, , sk(C) := C (C) (. (3)).

1.1.5. C X., h X ( ) - C, x + th C x C t 0. C, rec(C) (a(C)), ,

a(C) := rec(C) :=

{(C x) : x C, R, > 0}.

(1) rec(C) y X, C + y C.

, C + y C. C + ny = (C + y) + (n 1)y C + (n 1)y . . . C,

. . x + ny C x C n N. C , - x + (n 1)y x + ny, C. C x + ty t 0, , y rec(C). .

(2) rec(C) . , rec(C) X, C + rec(C) C.

t 0 t rec(C) = rec(C). , x, y rec(C) 0 1, (1)

C + x+ (1 )y = (C + x) + (1 )(C + y) C + (1 )C C.

Copyright & A K-C

16 1.

(1), : x+(1)y C. , rec(C) . (1).

(3) rec(C) = C , C .

C , (C x) C x C 0. ,

C = C 0

{(C x) : x C, 0} = rec(C) C, . (2).

(4) , C, {y X : C + y = C} {y X :x+ ty C (x X, t R)}.

1.1.6. - . - - . , , , . CS(X) - X.

(1) (. 1.1.1 (1)).

, CS(X), , .

(2) (. 1.1.1 (3)).

: (C,D) C D CS(X) CS(Y ) CS(X Y ) .

(3) L(X,Y ) X Y . T (C) C CS(X) T L(X,Y ) , . . CS(Y ).

CS(T ) : C T (C) CS(X) CS(Y ) . ( CS(T ) .)

(4) C1+. . .+Cn := {x1+. . .+xn : xk Ck, k := 1, . . . , n} C1, . . . , Cn .

n : Xn X, n(x1, . . . , xn) :=

nk=1

xk.

1 , 2 +. C1+. . .+Cn = n(C1. . .Cn), (2) (3).

, , . + CS(X), . .

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1.1. 17

, , {0} CS(X). CS(T ) (2) (3) ( CS(X)CS(Y ) ).

(5) (. . 0 < < ) C := C := {x : x C}. , C , C. R+ {+} . ,

0 C := rec(C), 10 C := C := coneC;

0C := 0,10C := C := X (C CS(X)).

, C = C = 0 = . = = (0 ).

:

(C1 + C2) = C1 + C2,

(+ )C = C + C (0 , ). C1 C2.

(= ). , . , 0 < , < . := /( + ) := /( + ) + = 1. , C :

C = C + C =1

+ (C + C),

. , C

= 0 = . , rec(C1 + C2) rec(C1) + rec(C2),

rec(C) + C C, cone(C) rec(C) + cone(C),cone(C1 + C2) cone(C1) + cone(C2),

.(6) (C) , -

, . -, . . , , C C C C ,

C .

(7) (C) - (6) {

D : Pfin()},

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18 1.

D := co({C : }) . -

C , , D

x, x C. ,

co(

C

)=

Pfin()

{

C : 0,

= 1}.

, := {1, . . . , n} (. (4)):

co(C1 . . . Cn) ={

1C1 + . . .+ nCn : k 0, 1 + . . .+ n = 1}

=

={

n

( nk=1

kCk

): k 0, 1 + . . .+ n = 1

}.

(8) #

C1# . . .#Cn :={

(1 C1) . . . (n Cn) : k 0, 1 + . . .+ n = 1}.

, - (5) (:0C = rec(C)). C1# . . .#Cn C1, . . . , Cn.

- . , x y X , . , x = e y = e 0, 0 e X. = 0 = 0,

z :=(1+

1

)1e.

= 0, z := 0. z x y e . x y x# y. , X, -, . , 0 < < 1 C1 (1 )C2 x X, - x = x1 = (1)x2 , , x = x1 #x2 (xk Ck, k := 1, 2)., :

C0 := {x1 #x2 : xk Ck, k := 1, 2} =

01C1 (1 )C2.

, C0 . C0 C1 C2. , , ,

C1#C2 = C0 (rec(C1) C2) (C1 rec(C2)).

Copyright & A K-C

1.1. 19

, C1#C2 = C0, , , C1 C2 .

1.1.7. ( ) ( ).

C1 C2. C := C1#C2, C0 , 1.1.6 (8). , x, y C x y C. z := x + y, , > 0, + = 1. -, C0. 1, 2, 1, 2 xk, yk Ck, k = 1, 2,

x = 1x1 = 2x2, y = 1y1 = 2y2.

k := k + k (k := 1, 2) , x = y : 1 = 0 2 = 0.

z1 :=11

x1 +11

y1, z2 :=22

x2 +22

y2,

zk Ck. ,z := 1z1 + 2z2 1C1 2C2 C0.

, C0 . x

, y rec(C1) C2. 1 = 0,

1 := 1, 2 := 1 1,z1 := x1 +

1y, z2 :=

22

x2 +

2y.

, zk Ck, z = 1z1 + 2z2 1C1 2C2., y , x C1 rec(C2), :

z = (x+ (/)y) C1, z = (y + (/)x) C2. , z C. C.

1.1.8. , 1.1.6, -. , , , - . , - , .

1.1.6 (7, 8) 1.1.5 (3) , K1, . . . ,Kn

K1 + . . .+Kn = co(K1 . . . Kn);K1# . . .#Kn = K1 . . . Kn.

Copyright & A K-C

20 1.

K K K = lin(K) = aff(K) , K, K (K) -, K. K , X = K K, , , , , K (K) = {0}.

1.1.9. 1.1.4 (13, 57) - - M m -, H(M). , X , - , m dim(X) 1.1.4 (1, 3) dim(X) + 1 1.1.4 (57).

(1) X n. , () M X, () n M .

. x cone(M). 1.1.4 (3) x = 1x1 + . . . +mxm, 1, . . . , m R+ x1, . . . , xm M . , m > n k . x1, . . . , xm 1, . . . , m, 1x1 + . . .+mxm = 0, k . , , k < 0 k. := min{k/k : k < 0} k := k + k.

k 0,

. ,

x =m

k=1

kxk + m

k=1

kxk =m

k=1

kxk.

{x1, . . . , xm} xk, k = 0, - x . - x1, . . . , xm x .

(2) . X - n. , (, ) M X, (, ) n+ 1 M .

. x co(M), 1.1.4 (5) x = 1x1 + . . . + mxm, 1, . . . , m R+ 1 + . . . + m = 1. y := (x, 1) yk := (xk, 1)(k := 1, . . . ,m). Y := X R y = 1y1 + . . . + mym. , y co(M ), M := {(x, 1) : x M} X R. m > n+ 1 = dim(Y ), (1) 1, . . . , n+1 : {1, . . . , n + 1} {1, . . . ,m} , y =1y(1) + . . . +

n+1y(n+1). , x = 1x(1) +

. . .+ n+1x(n+1), 1 + . . .+ n+1 = 1.

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1.2. 21

(3) x0, x1, . . . , xn X - , 0, 1, . . . , n R 0x0 + 1x1 + . . . + nxn = 0 0 + 1 + . . . + n = 0 , 0 = 1 = . . . = n = 0. , x0, x1, . . . , xn x1 x0, . . . , xn x0. ( , xn+1 := x0.) , x1x0, . . . , xnx0 - , 1 k n x0 xk, . . . , xk1 xk, xk+1 xk, . . . , xn xk. - x0, x1, . . . , xn n- , x0, x1, . . . , xn . S x0, x1, . . . , xn, x S - x = 0x0 + 1x1 + . . . + nxn, 0 0, . . . , n 0 0 + 1 + . . . + n = 1. 0, . . . , n - x.

(4) M n- - m- , m n, M .

(2) (3).

1.2.

, . , - x X (x) Y . X P(Y ) - , , - X Y . F := {(x, y) : x (x)}, X Y , . , , := (X,Y, F ) , F X Y . -, . , F , X Y . , , , F , , - . , .

1.2.1. .(1) X Y , F -

X Y . := (X,Y, F ) ( F , ) X Y . dom() im()

dom() := {x X : (y Y )(x, y) F};

Copyright & A K-C

22 1.

im() := {y Y : (x X)(x, y) F}. U X, (U, V, F0), F0 := F (U Y ) U Y U |U U .

U (U) := im( U). (x) := ({x}), -, :

(x) = {y Y : (x, y) F}; dom() = {x X : (x) = };(U) =

{(x) : x U} = {y Y : (x U) y (x)}.

(2) Z Y Z. -

1 := {(y, x) Y X : (x, y) }; := {(x, z) X Z : (y Y )(x, y) (y, z) }.

1 Y X , X Z ( ; ). : XY Z XZ (x, y, z) (x, z).

= (( Z) (X )).

:

( )1 = 1 1;( )(M) = ((M)) (M X).

:

( ) =

(b,c)1(b) (c) = ( ).

: X Y f X Y , dom() =X :

(x, y1) (x, y2) y1 = y2.

(x) = {f(x)} (x X) f . f g , g f - gf .

(3) X Y . A X( )

(A) := {y Y : A {y} } = {y Y : 1(y) A}.

Copyright & A K-C

1.2. 23

, (A) y Y , - x A (x, y) . , (x) = (x)(x X),

(A) =xA

(x) =xA

(x) (A X).

, . (A) (A). , A = {x} (A) (x). 1 1(B) := 1(B) (B Y ).

:(a) A1 A2 X, (A1) (A2).(b) AB , B (A) A 1(B).() A X B Y , A 1((A)) B (1(B)).(d) (A) X,

(

A) =

(A);(e) A X B Y , (A) = (1((A))) 1(B) = 1((1(B))).

(4) . A X. A = 1(B) B Y , x X, A, y0 Y , 1(y0) A x / 1(y0). , A = 1((A)).

: , B . x X\A. x / A = 1(B), ({x}B)\ = . , y0 B , (x, y0) / . , x / 1(y0) = 1(y0). y0 B (a) (3) 1(y0) 1(B) =A. , y0 .

: , . , A = 1((A)) , , B = (A). - (b) (3) A 1((A)). x 1((A)). , x / A, y0 Y , . , (a) (b) (3), y0 (1(y0)) (A), 1(y0) 1((A)), x 1((A)).

1.2.2. X Y X Y . , -, P(X Y ). - - (. 1.1.2), -. -, , , , , (. 1.3.5 (3)). -: , ,

Copyright & A K-C

24 1.

( (. 1.3.4)). - -, , A,B R2 := A B (2), (4) (5).

(1) X Y -, A X,B X (, )

(A+ B) (A) + (B)., a, b X (, )

(a+ b) (a) + (b), -.

P(X Y ). (, ) = (0, 0) - A, B , . - y (a) z (b), a A b B . (a, y) + (b, z) . ,

y + z (a+ b) (A+ B). (a)+(b) (A+B) a A b B, .

, y+z (a+b), y (a),z (b) (, ) . (a, y) + (b, z) (, ), y z, P(X Y ).

(2) A-, X Y , C PB(X). (C) P(Y ).

- C P(X). (, ) (1)

(C) + (C) (C + C) (C).

(3) -, 1 -.(4) XY A- Y Z

B-. -. , P(X Y ) P(Y Z).

, (, ) u, v X (1) (u+ v) ((u) + (v)) ((u)) + ((v)),

, -. (5) X Y A-, M P(X),

H((M)) (HB(M)).

1.2.3. -, , . .

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1.2. 25

. - -, - . , , 1.1.6, -. , , x .

, - -.

() X Y :

(1)(

)(x) =

(x);

(2)(

)(x) =

(x);

(3) co(

)(x) =

Pfin()

{ k

kk(xk)},

x =k

kxk, xk X, k R+,k

k = 1.

(4) X Y .

X :=

X, Y :=

Y,

: ((x, y)) ((x), (y)) . (

) X Y

(

)(x) =

(x) (x X).

(5) T L(U,X), S L(V, Y ) U V , (S T )() X Y ,

(S T )()(x) = S((T1(x))) (x X).

, , L(U,X) L(V, Y ) , - U X V Y .

(6) ,

(x) = (x/) (x X).

Copyright & A K-C

26 1.

, 0(x/0) := (0 )(x) ()(x/) := ( )(x). 1.1.6 (5)

0(x/0) =

{(u+ x/) v 0, (u, v) } ;(x/) =

{(x/) > 0} .

-.

:(7) (1 + . . .+n)(x) =

{1(x1) + . . .+n(xn) : x = x1 + . . .+ xn};(8) (1# . . .#n)(x) =

{11(x/1) . . . nn(x/n)}, 1, . . . , n R+ , 1 + . . .+ n = 1.

1.2.4. . , (. 1.2.1). .

, 1, . . . ,n X Y . 1 . . .n . (x, y) 1 . . .n , y = y1 + . . . + yn, y1, . . . , yn Y , (x, yk) k := 1, . . . , n. ,

(1 . . . n)(x) = 1(x) + . . .+n(x) (x X). 1 . . .n

dom(1) . . . dom(n). 1+. . . . +. n. (x, y)

1+. . . . +. n , x = x1+ . . .+xn, xk X (xk, y) k k := 1, . . . , n.

(1+. . . . +. n)(x) ={

1(x1) . . . n(xn) : xk X,n

k=1

xk = x

}.

1+. . . . +. n dom(1) + . . . + dom(n). - :

(1 . . . n)1 = 11 +. . . . +. 1n ;(1+. . . . +. n)

1 = 11 . . . 1n ., -

1.1.6 (1)(3). n , - (X Y )n Xn Y n. ,

n((x1, y1), . . . , (xn, yn)) := (x1, . . . , xn, y1, . . . , yn).

Copyright & A K-C

1.2. 27

: Xn Y n X Y

: (x1, . . . , xn, y1, . . . , yn) (1n

nk=1

xk,n

k=1

yk

).

(1 . . . n)(x) = (n

( nk=1

k

) (n(X) Y n)

).

, , n : x (x, . . . , x) X n(X) :={(x, . . . , x) Xn : x X} Xn. . 1.1.7 (, ), .

() - () . - -.

1.2.5. X Y Y Z. -

:=

+=10, 0

( ) ( )

. , X Z. (x, z) X Z , , R+, + = 1, y Y , (x, y) (y, z) . , 0 = rec() 0 = rec(). (1/M) = 0, 0(1/0M) := rec()(M) (. 1.2.3 (6)). :

=

+=10, 0yim()

()1(1y

)

(1y

);

(x) =

+=10, 0

(

(x

)).

, , ()1 = 11. , = .

. x1, x2 X 1, 2 R, 1 = 0, 2 = 0, 1 + 2 = 1. -

, (x). , , , R+ , + = 1 = + .

Copyright & A K-C

28 1.

1 := 1+ 2 2 := 1 + 2. , 1 = 0 2 = 0. 1 + 2 = 1 :

A(, , , ) := 1(

(x1

))+ 2

(

(x2

))

2(1

2(x1

)+

2

2(x2

))

2(12

(1x1 + 2x2

1

)) (1x1 + 2x2).

1 = 0, = = 0.

A(, , , ) = 1(rec()(x1)) + 2(rec()(x2)) (1 rec()(x1) + 2 rec()(x2))

(rec()(1x1 + 2x2)) (1x1 + 2x2).

= = 0. , , , , A(, , , ) (1x1 + 2x2).

A(, , , ) = 1((x1)) + 2((x2))

. 1.2.6. C X

H(C) X R, - C. :

H(C) := {(x, t) X R+ : x tC}.

(1) , - .

, (X {1}) H(C) = C {1}. H(C) C. , , C -. x, y X. s H(C)(x) t H(C)(y). 1.1.6 (5) sC + tC = (s+ t)C. , x+ y (s+ t)C s + t H(C)(x + y). , H(C)(x + y) H(C)(x) + H(C)(y). - H(C) . 1.2.2 (1) , H(C) .

(2) C :

co(H(C)) = cone(H(C)) = H(co(C)).

, C1 C2 H(C1) H(C2). (1) - , co(H(C)) , H(co(C)) . -, co(H(C)) H(co(C)). x co(C), > 0, x = (1x1+. . .+nxn)

Copyright & A K-C

1.2. 29

x1, . . . , xn C 1, . . . , n , 1 + . . .+ n = 1. xk C, (xk, ) H(C).

(x, ) =n

k=1

k(xk, ) co(H(C)),

. , ,

.

1.2.7. CSeg(X) Cone(X) - X. H : C H(C) CSeg(X) Cone(XR). CSeg(X) - H . :

H(C1 . . . Cn) = H(C1) . . . H(Cn);H(co(C1 . . . Cn)) = H(C1) + . . .+H(Cn);H(C1 + . . .+ Cn) = H(C1)+. . . . +. H(Cn);

H(C1# . . .#Cn) = H(C1) . . . H(Cn).

1.2.8. , - , , , - , . . . .

A B X. a A A B, b B \ {a} > 0, a + t(b a) A 0 < t < . A, , coreB(A) A B. a coreB(A) , , a b B, - a, A. core(A) := coreX(A) - A , , A. , A , 0 core(A). A ri(A) := coreaff(A)(A). , A , X =

{nA : n := 1, 2, . . . }; . ., , A X.

(1) X Y . A X B Y . V X

coreB((A)) (coreA(V )) coreB((V )).

Copyright & A K-C

30 1.

y , y (x) x coreA(V ).

0 := (x, y), A0 := A x, V0 := V x, B0 := B y., , 0(V0) = (V ) y 0(A0) = (A) y.

0 coreA0(V0), 0 coreB0(0(A0)). , , 0(V0) B0. , b B0, > 0 , b 0(A0). (a, b) 0 a A0. V0 A0, 0 < < 1, a V0. :

(a, b) = (a, b) + (1 )(0, 0) 0. b 0(x) 0(V0), -.

(2) 0 (0) im() , .

(3) C X , core(C) = C. C X, - X \C . , C , core(X \ C) = X \ C.

(4) C X , rec(C) ={C : > 0}. , - .

, C = X, . K :={C : > 0}. K rec(C) . k K , k + C C + C (1 + )C > 0. x X \ C, x core(X \ C). , 0 < < 1, (1 )x = x+ (x+ 0) X \ C. , (1 + )(1 ) < 1. x (1+)C, x = (1+)c, c C, (1)x = (1+)(1)c C (X \C), . , C =

{(1+)C : > 0} C+k C. 1.1.5 (1) , k rec(C).

1.3.

- -. .

1.3.1. - E, + := .

Copyright & A K-C

1.3. 31

E E := E {,+}. , + , E, , - E E, E. , + := inf := sup. , E, E. :

x+ y := x+ y := inf{x + y : x x, y y; x, y E} ( 0);() := () := ; ()() := ()() := + ( > 0).

, x := +x := x E {};0() := ()0 := 0, (0,0, x+,+x, x E) +. , . .

, E , . - , . . (x) = ()x, .

1.3.2. f : X E :(1) epi(f) := {(x, e) X E : e f(x)} -

;(2) x1, x2 X; y1, y2 E [0, 1] , f(xk) yk

(k := 1, 2),

f(x1 + (1 )x2) y1 + (1 )y2;

(3) x1, . . . , xn X 1 0, . . . , n 0 , 1 + . . .+ n = 1,

f(1x1 + . . .+ nxn) 1f(x1) + . . .+ nf(xn).

(1) (3): , epi(f) . f(xk) = k {1, . . . , n}, . x1, . . . , xn dom(f) := {x X : f(x) 0 1.2.3 (5) (1)(x) =1(x). , (a) 1 = . ,(x) = 1(x) (x) = (x). p(x) = p(x). x := 0 := 2, p(0) = 2p(0). , (0, 0) epi(p) , 0 dom(p). , p(0) = 0. p 1.3.2.

(b) (c): , - p,

p(x+ y) = p(12(2x) +

12(2y)

) 1

2p(2x) +

12p(2y) = p(x) + p(y).

(c) (d): .(d) (a): 1.3.2, (c) epi(p). (x, y) epi(p)

> 0, p(x) p(x) y. , (x, y) epi(p). , x = y = 0 = = 0 p(0) 0, . . (0, 0) epi(p).

(3) X Y . A : X Y (), A () X Y (. 1.2.2).

A : X Y (), A(1x1 + 2x2) = 1A(x1) + 2A(x2)

Copyright & A K-C

34 1.

x1, x2 X 1, 2 R, 1+2 = 1 (, x1, x2 X 1, 2 R).

X Y , , L(X,Y ) (. 1.1.6 (3)). 1.1.4 (2) - . T L(X,Y ) y Y , T y : x Tx + y (x X) . , A : X Y , (T, y), T L(X,Y ) y Y , , A = T y. , - . A(x), , : Ax.

, - .

1.3.5. X , E K- X E. f := inf , -

f(x) := inf (x) := inf{e E : e (x)} (x X), , - g : X E, epi(g) . ,dom(f) = dom(). (0) , - f .

x, y X, 0 0 , + = 1. (x) = (y) = , f . , (x) (y) ., (. 1.2.2 (1) 1.3.1),

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

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

, (0) . (x) = (x) x X > 0 (. 1.3.4 (2)). ,

f(x) = inf (x) = inf (x) = inf (x) = f(x).

, (0, 0) , f(0) 0 0 dom(f). ,f(0) = f(2 0) = 2f(0), f(0) E, f(0) = 0. , - f : X E , 1.3.4 (2)f .

1.3.6. 1.3.5 . - ,

Copyright & A K-C

1.3. 35

. - .

(1) . A E B E inf(A B) inf A inf B, . A := [a,+) := {e : e a} B := [b,+), inf(A B) =a b. 1.2.3 (1), .

f : X E ( ) - f := sup{f : },

f(x) = sup{f(x) : } (x X),

, epi(f) ={epi(f) : }.

, , dom(f)={dom(f) : }. -

:= {1, . . . , n}

f1 . . . fn := sup{f1, . . . , fn} = sup{fk : k := 1, . . . , n}.

(2) . , :={epi(f) : } , , inf . - f := inf (f) 1.2.3 (2) :

f(x) = inf{f(x) : } (x X).

() (f) - , . . , , f, f f , (-) .

(dom(f))

dom(f) =

dom(f).

(3) . f X E. X :=

X E :=

E. :

(X E)

X E (. 1.2.3 (4)). :=

( epi(f)

). ,

f :=

f := inf : X E

epi(f) = dom(f) =

dom(f) (. 1.2.3 (4)). ,

f(x) E, f(x) : f(x) ( )

Copyright & A K-C

36 1.

x dom(f), x : x, , f(x) = + , f(x) = + f(x) = , x dom(f) f(x) = . := {1, . . . , n} f1 . . . fn :=

nk=1 fk.

1.3.7. -.

(1) . f1, . . . , fn :X E := epi(f1)+. . .+epi(fn). inf - ( inf- +-(. 1.3.10)) f1, . . . , fn.

nk=1

fk := f1 . . . fn := inf .

1.2.3 (7), (n

k=1

fk

)(x) = inf

{n

k=1

fk(xk) : x1, . . . , xn X,n

k=1

xk = x

}.

. := R({0}), f = f = f , . . . ( , - - .) , dom(f1 . . . fn) = dom(f1) + . . .+ dom(fn).

() - () .

(2) . f := X E , 0 := epi(f). f := inf , epi(f) = epi(f) 0. :

f : x f(x/) (x X, > 0);0f(x/0) := f0(x) = sup

udom(f)(f(u+ x) f(u)) (x X).

f , f = f 0.1.3.8. - ,

.

(1) .

:= co(

{epi(f) : }).

f := co((f)) := inf (f). :=

Copyright & A K-C

1.3. 37

{1, . . . , n} f = co(f1, . . . , fn). 1.2.3 (3), (x, e) , xk X k R+,

x =n

k=1

kxk, e n

k=1

kfk(xk),n

k=1

k = 1.

,

f(x) = inf

{n

k=1

kfk(xk) : xk X, k R+,n

k=1

k = 1,n

k=1

kxk = x

}.

(f). co() . 1 2 , co(1) co(2). (co()), Pfin() , . 1.3.6 (2) inf{co()} = inf , := {epi(co())}. , inf = inf ,

f(x) = infPfin()

inf{

kkfk(xk)

},

{(xk, k)k : xk X, k R+,

k

k = 1,k

kxk = x}.

f co(

dom(f)).

, f1, . . . , fn ,

co

(n

k=1

epi(fk)

)= epi(f1) + . . .+ epi(fn).

,

co(f1, . . . , fn) := co({f1, . . . , fn}) = f1 . . . fn.

p : X E ( ), -

p := inf

{

p : Pfin()}

= co({p : }).

(2) .

:= epi(f1)# . . .#epi(fn).

Copyright & A K-C

38 1.

1.2.3 (8), 1.3.6 1.3.7 (2), f := inf

f(x) = inf

{(f11)(x) . . . (fnn)(x) : k R+,

nk=1

k = 1

}=

= inf

{1f1

(x

1

) . . . nfn

(x

n

): k R+,

nk=1

k = 1

}.

(3) . := epi(f1) . . . epi(fn). 1.2.4 (x) = f1(x) + . . . + fn(x) + E+, x dom(f1) . . . dom(fn) (x) = . ,

f1 + . . .+ fn := inf : x f1(x) + . . .+ fn(x) (x X).

, f1 + . . . + fn ,

epi(f1 + . . .+ fn) = epi(f1) . . . epi(fn),dom(f1 + . . .+ fn) = dom(f1) . . . dom(fn).

(4) . :=epi(f1)+. . . . +. epi(fn). ,

inf

(n

k=1

[fk(xk), +))

= f1(x1) . . . fn(xn).

, (. 1.2.4)

(x) ={ n

k=1

[fk(xk), +) : x1 + . . .+ xn = x}.

f := inf :

f(x) = inf

{f1(x1) . . . fn(xn) : xk X,

nk=1

xk = x

}.

f ( ) - f1, . . . , fn f1# . . .#fn. ,

dom(f1# . . .#fn) = dom(f1) + . . .+ dom(fn).

-.

Copyright & A K-C

1.3. 39

1.3.9. (1) . X Y , h : Y E . := epi(h) , h :=inf X E

h : x inf{h(y) : y (x)} (x X)., dom(h) = 1(dom(h)). (, - ), h = h . A := 1,

h : x inf{h(y) : Ay = x} (x X)., Y = epi(f)

f : X Y . (hf) :=epi(f)h := h f h,

(hf)(x) = inf{h(y) : y Y, y f(x)} (x X)., . ,

, -. , f Y , h , h(+) := +,

(hf)(x) = h(f(x)) (x X). h := Y E - Y h(+) := + h() := .

(2) . h , (1), := epi(h). , R+, + = 1,

inf{(( epi(h)) ())(x)} = inf{h

(y

): y

(x

)}.

, 1.3.8 (4), ,

(inf )(x) = inf+=1, 0

inf{h

(y

): y

(x

)}=

= inf{(h)(y) : y (x/), , 0, + = 1}., = epi(f) f := X

Y , h .

(inf )(x) = inf{(h) (f)(x) : , 0, + = 1} =

= inf{h

(

f(x

)): , 0, + = 1

}.

1.3.10. , , +- -. f1 :X1 X E f2 : X X2 E.

Copyright & A K-C

40 1.

epi(f1,X2) := {(x1, x, x2, e) W : f1(x1, x) e},epi(X1, f2) := {(x1, x, x2, e) W : f2(x, x2) e},

W := X1XX2E. X1X2 E

(x1, x2) :=xX

(epi(f1,X2) epi(X1, f2))(x1, x, x2);

(x1, x2) :=xX

(epi(f1,X2) epi(X1, f2))(x1, x, x2).

f2 f1 := inf , f2 f1 := inf .

,

dom(f2 f1) = dom(f2 f1) = dom(f2) dom(f1).

f2 f1, . . +- f2 f1, f2 f1.

:(1) +- -

(f2 f1)(x1, x2) = infxX

(f1(x1, x) + f2(x, x2)),

(f2 f1)(x1, x2) = infxX

(f1(x1, x) f2(x, x2));

(2) f1 f2 , f2 f1 f2 f1 ;

(3) , . .

f2 f1 = (f1 f2) ,f2 f1 = (f1 f2) ,

(f1 f2) f3 = f1 (f2 f3),(f1 f2) f3 = f1 (f2 f3),

: (x1, x2) (x2, x1) (x1, x2) X1X2 ( , f1 f2 f2 f3 );

(4) h (= o-) E K- F , f1, f2, f2 f1 f2 f1,

h (f2 f1) = (h f2) (h f1),h (f2 f1) = (h f2) (h f1);

Copyright & A K-C

1.3. 41

(5) - +-

f2 f1 = sup{( f2) ( f1) : , L+(E), + = IE},

, , L+(E) E, IE - E.

(1): .

(2): 1.3.5. . X Y Z (x) := {(x, y) : y Y }, X Z. , x, y X 0 < < 1

(x) + (1 )(y) =uY

(x, u) +vY

(1 )(y, v) =

=

u,vY((x, u) + (1 )(y, v))

u,vY(x+ (1 )y, u+ (1 )v) (x+ (1 )y).

(3): . - - 1.3.1, .

(4): h ,

h(f1(x1, x) + f2(x, x2)) = h f1(x1, x) + h f2(x, x2);h(f1(x1, x) f2(x, x2)) = h f1(x1, x) h f2(x, x2).

, h(inf A) = inf h(A) A E. .

(5): , - 1.3.11.

1.3.11. . f : X E , g : E F K- F , dom(g) = E.

infxdom(f)

supg

f(x) = supg

infxdom(f)

f(x).

4 (. 4.1.10 (2)). - , , , g := {A L(E,F ) : (e E) Ae g(e)} g(. 1.4.11).

Copyright & A K-C

42 1.

1.4.

. .

1.4.1. X Y . X Y , x, x1, x2 X R, > 0, :

(1) (x) Y ;(2) 0 (0);(3) (x) = (x);(4) (x1 + x2) (x1) + (x2).

, dom() = X (x) = (x) x X.1.4.2. .(1) X Y , Y Z.

. , . , () X Y () .

(2) X Y . XY , (x) := {Tx : T }, .

(3) C Y , T X Y p X.

(x) := p(x)C + Tx (x X).

:(a) p ,

;(b) p 0 C,

, , C p(x) = p(x) x X.

(4) () X Y

: x co((x) (x)) (x X)

() .(5) X Y , K X, K

.1.4.3. ,

. , F - , [a, b]+[c, d] = [a+c, b+d] a, b, c, d F a b, c d. , , [a, b] := {y F : a y b}

Copyright & A K-C

1.4. 43

F . , - [0, a + b] =[0, a] + [0, b] a, b F+.

X , F - . p, q : X F - , (p+ q)(x) 0 x X,

:= {(x, f) X F : q(x) f p(x)}, := {(x, f) X F : f p(x)}

. , q(x) = p(x) x X.

, , (1) 1.4.1. (2) (3) p q. u, v X, p q

(u+ v) = [q(u+ v), p(u+ v)] [q(u) q(v), p(u) + p(v)].

,

(u+ v) [q(u), p(u)] + [q(v), p(v)] = (u) + (v),

. . (4). , . - [q(x), p(x)] [p(x), q(x)], q(x) = p(x). .

1.4.4. E Y . X Y E -, (x) E x X.

. E Y

T (E ) := {y + C : y Y, R, C E }.

E C Y , T ({C}) T (C). F , I (F ) , . . I (F ) :={[a, b] : a, b F, a b}. , 1.4.2 (3) 1.4.3 T (C)- I (F )- .

1.4.5. (1) F - F+. , F - . I (F )- XF dom() = X p, q : X F , (x) = [q(x), p(x)] x X.

F0 := F+ (F+) F1 - F0. F - [a, b], a, b F1. , [a, b] = [a, b]

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44 1.

a, b F1, a a a a, a a F0 F1 = {0}, a = a. b = b. , x X a b F1 , (x) = [a, b]. p(x) := b q(x) := a. 2(0) = (2 0) = (0), [q(0), p(0)] [2q(0), 2p(0)] . , p(0) 0 q(0) 0. 0 (0) p(x) 0 q(x) 0. , [q(0), p(0)] F0, , p(0) = q(0) = 0. p q 1.4.1 (3) ., u, v X, , 1.4.1 (4) ,

[q(u+ v), p(u+ v)] = (u+ v) (u) + (v) == [q(u), p(u)] + [q(v), p(v)] = [q(u) q(v), p(u) + p(v)],

p q. (2) F -

, :(a) I (F )- X F ;(b) p : X F , (x) =

[p(x), p(x)] x X.1.4.6. E , -

E . , E , () - E0 E . T (C) ( T +(C) := {y + C : y Y, R+}) C Y , , C - ( ).

1.4.7. C Y - . C , . . y0 C , C y0 . C . , , C -, . . rec(C) = {0}, y0 .

{y + C : y C} , - y1+ y2 (y1+C) (y2+C) y1, y2 C. - . , y0 , y C y0 21(C+ y). y0 y C y0, y0 C C y0, Cy0 = (Cy0). Cy0 , y0 C. , y0 C. - , - .

, rec(C) = {0}, y1 y2 C. yk C = C yk (k := 1, 2). -

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, 2(y1 y2) + C = C. 1.1.5 (4) , y1 y2 rec(C) = {0}. y1 = y2.

1.4.8. , F I (F ) -. . , - F , a1, a2, b1 b2 F , ak bl (k, l := 1, 2), c F , ak c bl (k, l := 1, 2).

(1) - , - .

F - , z, u, v F+ , z u + v. a1 := 0, a2 := z v, b1 := u, b2 := z, , ak bl, k, l := 1, 2., c F ak c bl k, l := 1, 2. z1 := c z2 := z c z, . . z1 [0, u], z2 [0, v] z = z1 + z2. , F - , ak, bl F , ak bl k, l := 1, 2. u1 := b1 a1, u2 := b2 a2, v1 := b1 a2 v2 := b2 a1. uk, vk 0 (k := 1, 2) u1 + u2 = v1 + v2. - u1 = t11 + t12 t1k [0, vk] (k := 1, 2). t2k := vkt1k, t2k [0, vk] t21+t22 = u2, , t1k+t2k = vk (k := 1, 2). , b2 t22 = b1 t11 = a1 + t12 = a2 + t21, c ak c bl k, l := 1, 2.

(2) F - . I (F ) , F .

, I (F ) . - A F - E := {[a, b] : a A, b F, b A}. E . E . c , , c = supA. , ([a, b]) - , , a b , . a := sup{a : } b := inf{b : }. , a b [a, b]

{[a, b] : }. 1.4.9. . ,

, X Y () E Y . , E - , , +-, T (E ) , . . , C1, C2 T (E ) C1 + C2 T (E ).

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46 1.

X Y . T : X Y - , Tx (x) x X. . T0 Y , X0 X. T0 |X0 X0, . . T0 (|X0), T - T0 X, T . , X, - X0 X, T (E )- T0 (|X0). , (Y,E ) - .

1.4.10. . Y E - Y . (Y,E ) , E .

: , , T (E ) = E . X Y . , X0 = X, . x1 X \X0 - X1 X, x x := x+ x1, x X0, R. , T0 X0 Y , X0, T1 X1, T1 ( X1). , - , y1 := T1x1. x X y1+T0x = T1x1+T1x (x1+x) y1 T0x+(x1+x). , - , T0x + (x1 + x), x X0, y1. - . , - y1

y1

{T0x+(x1 + x) : x X0}

T1x1 := y1. , T1 : X1 Y , x := x1 + x, x X, R, T1x := y1 + T0x, . , y1 = 0

T1x = (y1 + T0(x/))

(T0(x/) + (x1 + x/) + T0(x/)) = (x1 + x) = (x).

, T1 (X1). , E

. , , Cx := T0x + (x1 + x) E x X0, , Cx .

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u, v X0. , 0 T0(u v) + (u v) = T0(u v) + (u+ x1 (x1 + v)) T0u+(x1 + u) + T0v (x1 + v) = Cu Cv.

, , Cu Cv = . u v, (Cx)xX0 . , T0 X0 .

- . A - (X , T ) , X X, X0, T L(X , Y ) T0, T ( X ). A - , (X , T ) (X , T ) X X T = T X . , , - (A,) (= ), . . A (A,) . , (A,) (X, T ). , X = X, T X (X, T ). , T .

: , (Y,E ) . (C) E . X R, . . x X , x : R { : x := x() = 0} .

X0 :={x X :

x = 0}.

X Y : x

xC (x X).

, ., x X0. -

x+ =

x

. , -

(x), ,

x = x+ ,

x = x (, ).

( , -, - .)

(x) =

x+ C

x C (x X),

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48 1.

(x) =

xC

xC =,

x(C C) (x X0).

(C) C C = , , 0 (x) x X0. , X0. T , X. -, T L(X,Y ), Tx (x) x X.

e X , e() = 0 = e() = 1. Te = Te , . , , (e) = C . , (C) Te, E .

1.4.11. , , .

p , X E, . . dom(p) = X. ( ) p p - X E, p, . .

p := {T L(X,E) : (x X) Tx p(x)},

L(X,E) X E. p - p. , X0 X, T0 : X0 E , T0x p(x) x X0. - X, X0, T0 p, T p, T0 X0 X, , E .

1.4.10 ,, , - -. , , .

1.4.12. .

(1) E - , p1, . . . , pn, X E,

(p1 + . . .+ pn) = p1 + . . .+ pn.

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. T (p1 + + pn) P T0, Xn n(X) Xn ,

P(x1, . . . , xn) := p1(x1) + . . .+ pn(xn),

T0(x, . . . , x) := Tx (x, x1, . . . , xn X). P , T0 T0z P(z) z n(X). T : Xn E , T P T n(X) T0. Tkx := T (0, . . . , 0, x, 0, . . . , 0), x k. Tk X E T = T1 + . . .+ Tn. ,

Tkx P(0, . . . , 0, x, 0, . . . , 0) = p1(0) + . . .+ pk(x) + . . .+ pn(0) = pk(x);

. . Tk pk. (2) E , -

E , E . E ( , ).

, E - . pk : R E,

pk : t t+yk (t R, k := 0, 1, 2), y0, y1, y2 E+ y0 = y1+y2. p0 = p1+p2 (1), pk t ty (t R), y [0, yk].

E . z [0, y1 + y2], y1, y2 E+. U := {u E : u z, u y1}. U , z1 E+, z1 = supU . , z y2 U , zy2 z1. , z2 := zz1 z = z1+z2 zk [0, yk] (k := 1, 2).

1.4.10 1.4.5, 1.4.8 1.4.12 - .

1.4.13. . - , .

.(1) . -

. . . .

K-.

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50 1.

(2) . , , .

1.4.14. p -, X K- E.

(1) x0 X T X E, p x0, . . , Tx0 = p(x0) T p.

X0 = {x0 : R} T0 : X0 E T (x0) := p(x0). 0 T (x0) = p(x0) = p(x0). < 0,

T0(x0) = p(x0) = ||p(x0) p(||x0) = p(x0).

, T0 pX0. - T T0 X, p. T p Tx0 = T0x0 = p(x0), .

(2) - , . . -:

p(x) = sup{Tx : T p} (x X). , .

(1). (3) x X (p)(x) := {Tx : T p}

[p(x), p(x)]. (1).

T0: y0 [p(x), p(x)] T0(x0) := y0.

(4) Y T - Y X.

(p T ) = p T. S (p T ). , p(T (y))

Sy p(Ty), Ty = 0 Sy = 0. , ker(T ) ker(S), ker(R) := R1(0) R. , - X T = S X : T (Y ) E. U0 - U0x0 p(x0) x0 X0 := T (Y ). , - U L(X,E) U0, p. , U p U T = S,. . S p T . .

, , T X0 X, - . 1.4.14 (4)

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- .

(5) E, . . E E, 0 IE .

( p) = p.

p ( p) . T ( p), x X p(x) Tx p(x), . . T - im() , im() ., ker() E (. 2.5 (4)). - ker() im(). : im() ker() , , = Iim(). S := T , , S : X E . , S p S = T . T p.

1.4.15. .

(1) . X , X0 , . . X0 + X+ = X. T0 X0 K-E T X E.

x X x0 X0 , x x0. x0 X0 x0 x T0x0 T0x0. , T0x0 . ,

p(x) := inf{T0x0 : x0 X0, x0 x} (x X)

p : X E. , . ,

p = {T L+(X,E) : T X0 = T0},

. (2) T X

K- E. , X0 X - S0 : X0 E, S0x0 Tx0(x0 X+0 ). S : X E S0 , S T .

p(x) := T (x+) (x X). p : X E ,p = [0, T ] S0x0 S0(x+0 ) T (x+0 ) = p(x0) x0 X0. S S0, S p.

1.4.16. Y . , Y , X

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T0 : X0 Y , X0 X Y , T X, , T = T0. , 1.4.9. , C Y ,

X Y, : x kxC, k > 0. , T : X Y , T k. k := T0, T0 ( X0), - T T0 T = T0. , 1.4.10 , Y - , Y . , - , .

1.4.17. , - .

1.4.16. . A Y , Y Y . (y) := y A. y (y) Y - Z := (Y ) l(A) A. K- -, D := [e, e], e : A R , . - z 1(z) (z Z) P : l(A) Y , P = 1. , P := P l(A) Z P = 1. , P (D) D, , 1.4.7 1.4.8 (2), P (D) . P (D) - Z, , Y .

, . .

1.4.18. . C Y . C () - , Y , :

(1) Y ;(2) e Y + y Y , -

[e, e] + y C rec(C) = Y + (Y +);

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1.5. 53

(3) Y . K-, , -

C(Q) Q C(Q), -, -.

1.4.19. .- , C(Q) Q.

1.5.

, - -, . , , , , , -.

1.5.1. V , 0. , V , +, . , V , : x y, x+z y+z, x, y, z V . Isa(V ) - V , . , h Isa(V ) ,

h : V V, h(0) = 0, h(x+ y) h(x) + h(y);x y h(x) h(y) (x, y V ).

Isa(V ) : (h1, h2) h1+h2 (h1, h2) h1 h2, (h1+h2)(v) := h1(v)+h2(v) h2 h1(v) := h2(h1(v)). (Isa(V ),+) . , , . . -:

h (h1 + h2) = h h1 + h h2,(h1 + h2) h = h1 h+ h2 h.

h1 h2 , h1(v) h2(v) v V . Isa(V ), + . , g, h1, h2 Isa(V ) h1 h2, h1g h2g gh1 gh2. , , Isa(V )

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54 1.

. Isa(V ), -, Hom+(V ). , Hom+(V ) Isa(V ). .

K- E. Lr(E) - (= ) (-) E. , - Lr(E) K-. , K-, - (. 2.2). - Lr(E) . Orth(E) Lr(E), IE , . . Orth(E) := {IE}dd, Ad := {b : (a A) |b| |a| = 0} A. , N K- F (z F ,y N , |z| |y| z N) ( U N , F , N , . . supF U N) F . , Lr(E), Orth(E) - (f -). (. 2).

, A := Orth(E) E. Inv+(A) A. , Inv+(A), 1 0. , - V . , V A-- , : A+ Isa(V ) , (Inv+(A)) Hom+(V ); () IV , , :

(1) ()(u v) = ()(u) ()(v) ( Inv+(A), u, v V );(2) u+ v w = (u+ v) (u+ w) (u, v, w V ).

u := ()(u). , , V ( ) , V ( -) A- . . h V A- , h(u+v) = h(u)+h(v) , A+ u, v V.

1.5.2. . X , E K-. Sbl(X,E) - , X E. 1.3.8 (3). A := Orth(E). E A- (, p) p( A+, p Sbl(X,E)) p : x (p(x)) (x X), (+) := + .

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Sbl(X,E) , p q , p(x) q(x) x X. Sbl(X,E) - Sbl(X,E) .

Sbl(X,E) Sbl(X,E) A- . - , -.

1.5.3. - . U L(X,E) - -, S, T U - , A+, + = IE , - S + T U . , - -. U L(X,E) - op(U ), - U . op(U ) - U .

(1) - op(U ) U L(X,E) :

op(U ) =

{n

k=1

k Tk : T1, . . . , Tn U , 1, . . . , n A+,n

k=1

k = IE , n N}.

U0 . , U0 - , U , op(U ) U0. , - U - - , T1, . . . , Tn U 1, . . . , n A+

nk=1 k = IE n

k=1 k Tk U . , - n N, n 2. S := n+1k=1 k Tk, T1, . . . , Tn+1 U 1, . . . , n+1 A+,

n+1k=1 k = IE . :=

nk=1 k ,

x X Sx (T1x . . . Tnx) + n+1 Tn+1.

,

S n+1 Tn+1 ( (T1 . . . Tn)). 2.1.7 (1), 1, . . . , n A+ ,

1 + . . .+ n = IE ; S n+1 Tn+1 =n

k=1

k Tk.

, S = T + n+1 Tn+1, T =n

k=1 k Tk T U . +n+1 = IE , U S U .

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56 1.

CS(X,E) -- L(X,E). U L(X,E) - , x X E {Tx : T U }. CSb(X,E) -, CS(X,E). CS(X,E) Inv+(A)

U + U := {T + T : T U , T U } (U ,U CS(X,E));U := { T : T U } ( Inv+(A), U CS(X,E)).

A+

U :=

TU

(T +

{(U T ) : Inv+(A), }

).

CSb(X,E) .(2) CS(X,E) CSb(X,E) -

A- . - - - , - .

1.5.4. . p : X Y E - , x X y Y

p(x, ) : v p(x, v), p(, y) : u p(u, y) (u X, v Y ).

BSbl(X,Y,E) , - X Y E. BSbl(X,Y,E) . p q, p(x, y) q(x, y) x X y Y . -, p (p). p(x, ) p(, y) - (p(x, )) (p(, y)) . 1.3.7 (1) , p . - A+ , 1.5.2, . . p(x, y) := p(x, y) p(x, y) < + p(x, y) := + . p p A+, p1 + p2 - p1 p2 .

BSbl(X,Y,E) , - ; BSbl(X,Y,E).

BSbl(X,Y,E) - A- . A-- BSbl(X,Y,E) A-- .

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1.5.5. . Fan(X,Y ) X Y , - . , , (x) (x) x X. (. 1.2.4).

()(x) = (x) (x X, 0).

() X Y , - Fan(X,Y ),

(x) = co(

{(x) : })

(x X).

, Y A-. A+

()(x) := (x) (x X), (x) 1.5.3 (1). Fan(X,Y ) .

1.5.6. . V A- . - A- [V ] A- : V [V ], [V ] [V ], - . h A- - V A- W , h A- [h] : [V ] [W ]. h , [h] .

, 0v = 0 v V , , v v (v V ) 0, A. , 0 + v = v = (0 + )v = 0v + v v 0v = 0. , A+, + , . . + Inv+(A).,

(v1 + v2) + (v1 + v2) = (+ )(v1 + v2) =

= (+ )v1 + (+ )v2 = v1 + v2 + (v1 + v2).

v1 + v2 v1 + v2 = (v1 + v2). V V

,

(v1, v2) + (w1, w2) := (v1 + w1, v2 + w2);

(v1, v2) := (+v1, +v2) + (v2, v1);

(v1, v2) (w1, w2) v1 + w2 v2 + w1,

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v1, v2, w1, w2 V A.

(v1, v2) (w1, w2) v1 + w2 = v2 + w1. , v := (v1, v2) w := (w1, w2) , v w w v. [V ] := V V/, :=V : V V [V ] , . . -. V V [V ] ,

(v) + (w) = (v + w), (v) = (v),

v w (v) (w) (v, w V V, A)., [V ] A-. (v) := V (v) := (v, 0) (v V ). A- V (V ), v, w V

(v, w) = ((v, 0) (w, 0)) = (v, 0) (w, 0) = (v) (w). , (V ) [V ]. , (v) (w) , v w. , (v1v2) (v1) (v2). , u,w V (u,w) (v1), (v2), u v1 + w u v2 + w. u v1 v2 + w (u,w) (v1 v2). , (v1 v2) = (v1) (v2). , (V ) . v, w [V ] . , v = v1 v2 w = w1 w2, v1, v2, w1, w2 (V ). , ,

v w = (v1 + w2) (v2 + w1) v2 w2., [V ] A-, (V ) A+-- (. . A+) , -.

h : V W. v1, v2 V

[h](V (v1, v2)) := W (h(v1), h(v2)).

V (v1, v2) = V (u1, u2), v1 + u2 = u1 + v2, , h(v1) + h(u2) =h(u1)+h(v2), W (h(v1), h(v2)) = W (h(u1), h(u2)). [h] : [V ] [W ]. - h, A- [h]. , v V

[h] V (v) = [h](V (v, 0)) = W (h(v), 0) = W (h(v))., [h] V = W h. [h].

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1.5.7. 1.5.6 A- Sbl(X,E), . [Sbl(X,E)] - X E. [V ], 1.5.6, , [Sbl(X,E)] Sbl(X,E) Sbl(X,E) EX , X E, . (p, q), p, q Sbl(X,E), - x p(x) q(x) (x X). [Sbl(X,E)] , EX , {p [Sbl(X,E)] : p(x) 0 (x X)}.

: Sbl(X,E) CS(X,E), - p p. - . , , sup :CS(X,E) Sbl(X,E)

sup(U ) : x sup{Tx : T U } (x X).

1.4.14 (2), sup Sbl(X,E). cop := sup. cop :

(a) cop cop = cop;(b) cop(U ) U (U CS(X,E));(c) cop A- ,

.

( ). - cop CSc(X,E). - (a),

CSc(X,E) = {U CS(X,E) : cop(U ) = U }.

sup A-- Sbl(X,E) CSc(X,E). 1.5.6 CSc(X,E), [CSc(X,E)] - . , .

. cop , , - A- [] [cop] - A- [Sbl(X,E)] [CSc(X,E)], []1 = [sup].

1.5.8. Fanb(X,L(Y,E)) X L(Y,E) , dom() = X (x) (. . ) x X. - Fanb(X,L(Y,E)) s() : XY E,

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60 1.

s() : (x, y) sup{Ty : T (x)}.

, s() . p : XY E. -

p X L(Y,E)

p : x p(x, ) := {T L(Y,E) : (y Y )Ty p(x, y)}.

p(x1, ) + p(x2, ) p(x1 + x2, ), 1.4.12 (1) :p(x1 + x2) p(x1) + p(x2). , p(x) = (p)(x) = p(x) > 0. , p . Fanc(X,L(Y,E)) Fanb(X,L(Y,E)) , (x) CSc(X,E) x X.

:

(1)

: BSbl(X,Y,E) Fanb(X,L(Y,E)),s : Fanb(X,L(Y,E)) BSbl(X,Y,E)

- ;

(2) cop := s - Fanb(X,L(Y,E)), Fanc(X,L(Y,E));

(3) s BSbl(X,Y,E);(4) s A- -

BSbl(X,Y,E) Fanc(X,L(Y,E));

(5) s , , - A- [] [s] - A- [BSbl(X,Y,E)] [Fanc(X,L(Y,E))], [] = [s]1.

1.5.9. , -, E := R . R-- . . , , . CSeg(X) - X. - 1.1.6. , C D C D.

CS+(X) := (CSeg(X), +, , ). R, > 0 C CSeg(X), C := 1C. ,

, 0 C cone(C) C.

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, , . . C D C D.

CS#(X) := (CSeg(X), #, , ). .

Sbl+(X) Sbl(X,R), . R, > 0, p Sbl(X,R), p 0, p := 1p. , , 0p := R(ker(p)),. . (0 p)(x) = 0, p(x) = 0, (0 p)(x) = + . - (. 1.3.8 (4)), p# q p, q Sbl(X,R)

(p# q)(x) = inf{p(x1) q(x2) : x = x1 + x2} (x X). Sbl+(X), , . . p q p q.,

Sbl#(X) := (Sbl+(X,R), #, , ).1.5.10. . CS+(X), CS#(X), Sbl+(X)

Sbl#(X) . , CSeg(X) Sbl+(X) -

. , .

CS+(X) Sbl+(X) . - # . - , , - . , 1.5.1 (2).

(1) (+ ) C = ( C)# ( C) (, R+). = 0 = 0,

(+ ) C = 1C =

01

C =

=

01

{ C cone(C)

}= ( C)# (0 C).

, = 0 = 0. = = 0 . -, = 0 = 0. 0 1

:=

1

1

+ ;

C# C =

01

C 1

C =

=

01C 1

+ C = (+ ) C.

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62 1.

, := /(+ )

C 1

C =

1+

C = (+ ) C.

, ( C)# ( C) (+ ) C.(2) (C1 #D) (C2 #D) = (C1 C2)#D.

CS#(X) , ,

(C1 #D) (C2 #D) = (C1 C2)#D. , Ck #D (C1 C2)#D (k := 1, 2). - x (C1 #D) (C2 #D). 0 , 1 , x C1 (1 )D x C2 (1 )D. , x (C1) (C2) (C1 C2), ,

x (C1 C2) (1 )D (C1 C2)#D. , . C1 C2 = (C1 C2), 0.

1.6.

, - . , - . - - .

1.6.1. X - E. ( -.) : X E+ (E-) , :

(1) x = 0 x = 0 (x X);(2) x = || x ( R, x X);(3) x+ y x + y (x, y X).

, (4) e1, e2 E+ x X,

x = e1 + e2, x1, x2 X , x = x1 + x2 xk = ek (k := 1, 2). , (4) e1, e2

E+, , , d-. (X, , E) ( (X,E), (X, ) X,

) ( E), - E- X. E

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1.6. 63

X. - (d-),

(X, )

(d-). () - .

1.6.2. x y = 0, x, y X x y. x 0 - z 0, 0 x z x (z x) = 0. z - z x x , z - x. , M := {x X : (y M) x y}, = M X, X. B(X) X, . , K B(X) , KK = X. h() K K . , X -, X . B(X) B(X) , , - . , X . X - , , (. 1.6.4 1.6.9). L E M X h(L) :=

{x X : x L} M := { x : x M}. ,

h(L) L X .(1) E0 := X

. B(X) - L h(L) B( X ) B(X).

, h - . h , - . , h

({0}) = {0} h

(X

) = X. , , h(L) = h(L) L B( X ). h(L) h(L) -. 0 = x h(L), x L y . x / h(L) , e x 0 < e L+. {e} y , x / h(L).

(2) x, y X , x+ y = x + y . , x y = 0 x x+ y + y ,

x (x+ y + y

) x x+ y x x+ y . y x+ y ,

x + y = x y x+ y ,

.

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64 1.

(3) e1, e2 E - x = x1 + x2 x1 = e1 x2 = e2.

, x1 = y1 = e1, x2 = y2 = e2 x = x1 + x2 = y1 + y2. x1 y1 y2 x2, x1 y1 x1 + y1 = 2e1 x2 y2 2e2. (2) 0 = (x1 y1) + (x2 y2) = x1 y1 + x2 y2 . , x1 = y1 x2 = y2.

(4) (1)X d- - L B(E0). h() K := h(L) K, x = h()x x X.

d- x X, x1, x2 X, x = x1 + x2, x1 = x x2 = x . , X K K. h() K K. h h()x K = h(E0), . . h()x E0. h()x = 0 h()x = h()x . h()x h()x. , (2),

x = (h()x + h()x

)= h()x .

, x = h()x = h()x . 1.6.3.

X B - , X, , :

:= = , = + , = IX (, B).

E0 := X , X d- -

. X . , B X h P(E0) B ,

b x = h(b)x(b P(E0), x X

).

L B(E0) X . , x / L x X. L, x . d- X, x0 X x0 = x L. , 1.6.2 (1, 4). K B(X) K K. P := {K : K B(X)}. , P . P(E0), K , K := h(E0). h. h P(E0) P. h 1.6.2 (4).

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1.6. 65

P(E0) P(X) := P x = x (x X, P(E0)).

1.6.4. , X . -, |x| |y| x y (x, y X).

(1) X E. X , -, B(X) B(X). -, X X.

, h(L) - X L B(E). 0 x h(L) 0 y h(L), 0 x y h(L) h(L) = {0}, xy x y . , xy = 0 , , x y 1.6.2, X.

d X, . .u d v |u| |v| = 0. h(L) d h(L). h(L) h(L)d, Ad := {x X : (a A)x d a}. h(L) = h(L)d L B(E).

, , x dh(L) x / h(L). x / L. 0 < e L, e x . X, u, v X, x = u + v, u = e v = x e. u h(L), x du, |x| |v|. x v = x e 0 < e 0. , x dh(L) x h(L). , h(L) = h(L)d. L L, , h(L) =h(L)d. h(L) B(X), . . B(X) B(X). , 1.6.2 (1), h(L) = h(L) = h(L)d. B(X) B(X). B(X) B(X) - , , B(X) B(X).

(2) X , (1), E -. P(X) P(X). , - X X.

(1) 1.6.3. 1.6.5. , (x)A bo- x X x =

bo-lim x, (e) E , inf e = 0 () A, x x e (). e E+ : > 0 () A , xx e (). , (x) br- ( e) x x = br-lim x. (x) bo- (br--), (x x)(,)AA bo- (br-) .

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66 1.

bo- (br-), bo- (br-) bo- (br--) .

(x) (y)A. A := Pfin() y :=

x. x := bo-lim y, , (x)

bo- x . x = bo-

x. X

, bo- o r bo br.

M X , M E, . . e E+, x e x M . X d-, bo- , .

. bo- , .

[129, 132]. bo- -

(, , ). - , . , - K-. , , bo-.

1.6.6. (bo-) - (X,E) - (Y,mE) ( - (Y, oE)) : X Y , bo- (Y,mE) ( bo- (Y, oE)), X, Y . oE E, mE - K- oE. , , E oE mE (. [129, 132]).

U X :

r(U) :={x = br-lim xn : (xn)nN U

},

d(U) :={x = bo-

x : (x) U},

o(U) :={x = o-lim

x : (x)A U

},

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1.6. 67

A , () P(X), X.

(1) . . (mX,mE) (X,E).

(2) . X -, , bo--.

(3) bo- X X X = rdX. X E0 := X

, X = oX.

(1)(3) . [129, 132]. 1.6.7. , (X,E) ,

.(1) x X -

(an) E+, (an) x (n N). (xn) X , n N n < m N

xn = an, x xn = x an, xm xn = am an.

bn := x an (n N), b0 := x . X (un) X (vn) X ,

x = u1 + v1, u1 = a1, v1 = b1,un+1 + vn+1 = vn, vn+1 = bn+1, un+1 = bn bn+1.

xn =n

k=1 uk. x = xn + vn :

xn n

k=1

uk =n

k=1

(bk1 bk) = b0 bn = an.

x xn + vn an + bn = x . , xn = an. m > n

xm xn =m

k=n+1

uk m

k=n+1

uk = am an xm xn ,

xm xn = am an. (2) (X,E) d-, (X,E) -

. , d- .

. , , [128]. y X, E(y) o- E, y , X(y) :=

{x X : x E(y)}. ,

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68 1. (X(y), E(y)

) . K X K(y) := K X(y),

y K K(y) . E(y) . , - (e), E(y), (x) X , x = e. x = bo-

x x = o-

x =

sup e E(y). , 1.6.2 (4) .

(3) (X,E) d- br-, E0 :=X

K- X = E+0 . (2) X . E

E. (2), E(y) . , e E(y)+ - en E(y)+, - e. (1), (xn) X , xn = en xm xn em en (m > n). , (xn) r- , br- X, x := br-lim xn X. , x = r-lim en = e, , E(y) = E(y). e - X . (e) X (n) E, ne ne, . . ne X . (xn) X x X , xn = ne e x = e. , E0 = X K- X = E+0 .

1.6.8. A Orth(E), A P(E). , - X E - A, X A-, :

(a) A X P(E) P(X), 1.6.3;

(b) ax = |a| x (a A, x X). , X ,

(a) (b), (c) B(X) B(X).

X d- E, E = X . A - Orth(E), P(E). - , X A:

(1) A , . . 11 +. . .+ kk, 1, . . . , k , 1, . . . , 2 - P(E);

(2) E , X br- A := Z (E);

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1.6. 69

(3) E -, X bo- A = Orth(E);(4) E -, X -

A = Orth(E). a A a = kk,

1, . . . , n R 1, . . . , n P(E). ax :=kkx. 1.6.2 (2) 1.6.3,

P(E) P(X),

ax =

kkx =

|k|k x = a x .

, (2) ( (3)), a A (- ) (an) A. (anx) X br- (bo--),

anx amx = |an am| x (r) 0 ( (o) 0).

, ax := br-lim anx (- ax := bo-lim anx). :

ax = br-lim anx = r-lim |an| x = a x ,ax = bo-lim anx = o-lim |an| x = a x .

. (4) - 1.6.4.

1.6.9. , (X,E) , - X E , . X : - 1.6.2, X. , , -, - .

(X,E) . x X n-,

nk=0 xk = 0

x0, x1, . . . , xn X+ , |x| =n

k=0 xk. - 1- . , , , . - E .

(1) X . x X n- ,

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70 1.

x0, x1, . . . , xn X+ x = x0+x1+. . .+xn {0, . . . , n} P(E) , kxk = 0 k = 0, 1, . . . , n.

, e0 . . . en = 0 , kek = 0(k := 0, 1, . . . , n) 0, . . . , n P(E). - ek := xk 1.6.2 (4), k xk = kxk = 0 kxk = 0.

(2) X - Orth(E). x X n- , x0, x1, . . . , xn X+ x = x0 + x1 + . . . + xn 0, 1, . . . , n , 0 + 1 + . . . + n = IF kxk = 0 k := 0, 1, . . . , n.

(1). . 0, 1, . . . , n -

. k ker(k) -, kxk = 0 (k := 0, 1, . . . , n).

nk=0 k = IE , n

k=1 ker(k) = {0}. ,

nk=1 ker(k) = 0. ,

nk=1 ker(k)

= E. ,

nk=1 k = IE . 0, 1, . . . , n

P(E) , n

k=1 k = IE k k (k := 0, 1, . . . , n). , kxk = 0 k := 0, 1, . . . , n. (1).

1.6.10. (1) X . n X n-.

x = u1+ . . .+un, u1, . . . , un - X. , uk 0 (k := 1, . . . , n). x0, x1, . . . , xn X, - x0 + x1 + . . .+ xn = u1 + . . .+ un. , uk,l X+ ,

uk =n

l=1

uk,l (k := 1, . . . , n); xl =n

k=0

uk,l (l := 0, 1, . . . , n).

vk,l E+ k := 1, . . . , n l := 1, . . . ,m, n-

nk=1

ml=1

vk,l jJ

v1,j(1) . . . vn,j(n),

J j : {1, . . . , n} {1, . . . ,m}. vk,l :=uk,l :

0 n

l=0

xl =n

l=0

nk=1

uk,l n

l=0

nk=1

uk,l jJ

uj(0),0 uj(1),1 . . . uj(n),n ,

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1.6. 71

J j : {0, 1, . . . ,m} {1, . . . ,m}. , {j(0), . . . , j(n)} , , j(r) =j(s) = m, 0 r, s n, r = s.

uj(0),0 uj(1),1 . . . uj(n),n um,r um,s = 0.

0 um,r xr, 0 um,s xs, um = um,0 + . . .+ um,n,

xr xs , um . (2) E ,

X d- E. x X n- x = x0 + x1 +. . .+ xn1 x0, . . . , xn1. E, (E) { x0 x1 . . . xn1

}, y := (x0) , z := (x x0) c (n 1)-. , y z x, . . y.

0 y1 y, 0 y2 y, y1 y2 = 0 y1 + y2 = y. , y1 y2 = 0. k { xk } := 01 . . . n1. n + 1 x1, . . . , xn1, y1, y2 ,

x1 + x2 + . . .+ xn1 + y1 + y2 = (x x0) + (x0) = x.

n- n-, x1 . . . xn1 y1 y2 = 0, , e = 0, e := ( x1 . . . xn1 y1 y2 ). e = 0, e y1 x0 = 0, , e = 0. , ( y1 y2 ) = 0, ( x0 . . . xn1 ). y1 y2 = ( y1 y2 ) x0 = 0. , y x.

, z = z1 + . . .+ zn zk zl (k = l). x x0, z1, . . . , zn n + 1 , x n-, x0 z1 . . . zn = 0. e = 0 ( x0 z1 . . . zn ) = 0. e = 0. , e = 0 ,, ( z1 . . . zn ) = 0, z1 . . . zn = 0.

(3) E , X d- E. -, n- x X x = x1 + . . .+ xn x = y1 + . . .+ ym (n,m N), {x1, . . . , xn} {y1, . . . , ym} . l := 1, . . . ,m 1,l, . . . , n,l P( yl

) , yl =n

k=1 k,lxk (l := 1, . . . ,m).

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72 1.

x 0. , - xk,l X (k := 1, . . . , n; l := 1, . . . ,m), xk =

ml=1 xk,l (k := 1, . . . , n) yl =

nk=1 xk,l (l := 1, . . . ,m). xk,l

0 xk,l xk yl. yl - , xk,l xj,l = 0 (j = l). k,l xk,l

. , 1,l, . . . , n,l - . 1.6.2 (2)

k,lxk xk,l = k,l xk xk,l == k,l( xk,1 + . . .+ xk,l1 + xk,l+1 + . . .+ xk,m ) = 0,

, k,lxl = xk,l. yl =nk=1 k,lxk. yl =

nk=1 k,l xk ,

nk=1 k,l yl =

nk=1 k,l xl = yl .

(k,l)nk=1 yl

l := 1, . . . ,m. x X |x| = |x1|+ . . .+ |xn| = |y1|+ . . .+ |ym|.

|yl| =n

k=1 k,l|xk|(l := 1, . . . ,m).

y+l n

k=1

k,lxk =

nk=1

k,lx+k yl .

y+l n

k=1 k,lxk y

l

nk=1 k,lx

+k

: y+l =n

k=1 k,lx+k y

l =

nk=1 k,lx

k . , yl =n

k=1 k,lxk. 1.6.11. . E

X d- E. n- x X - n , . . x = x1 + . . . + xn, x1, . . . , xn X. 1.6.10 (3).

n. n = 1 -. , n 1. P - E, : P n x0, x1, . . . , xn1 X,

n1i=0 xi = x

(E) ( x0 x1 . . . xn1 ). 1.6.10 (2), x, P, ,

x , y := xx (n1)-, x = x, x y x. (x)P , X d-, y :=

{x : P} z := x y. , y z x.

y = x P, , := supP, y = y y = 0. y = y1+y2, y1y2 = 0, y1+y2 = x

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1.6. 73

P ( y1 y2 ) = 0, x . ( y1 y2 ) = 0, ( y1 y2 ) y = 0, y1 y2 = 0. , z = z0 + z1 + . . .+ zn1 z0, z1, . . . , zn1 X+, e := z0 . . . zn1 . P z = y, , z (n 1)- e = 0. x = (z0 + y) + z1 + . . . + zn1 , P,

( z0 . . . zn1 ) ( z0 + y z1 . . . zn1 ) = 0. ,

n1i=0 zi = 0. , , y -

, z (n 1)- x = y + z, - x. n-- x |x| n- - |x| = y1 + . . . + yn, yk . x+ = u1 + . . . + un, x = v1 + . . . + vn, uk + vk = yk (uk, vk X+; k := 1, . . . , n). xk := uk vk , yk = |xk|. ,, x = x1 + . . .+ xn.

1.6.12. . X , E K-. p : X E , p(x) = p(x) x X. p - Z (p) L(X,E) : Z (p) Orth(E) :

Z (p) := {T L(X,E) : ( Orth(E))T ( p)},T := inf{ Orth(E)+ : T ( p)} (T Z (p)).

. (Z (p), ,Orth(E)) -, p , . . p(x) = 0 x = 0.

, - 1.6.1 (2, 3). T = 0, 0 < R () P(E) () Orth(E) , IE T (p) . , |Tx| p(x) p(x), , |Tx| p(x). p , T = 0.

E , T (p) , T ( p). , T ( p) 0 Orth(E), Orth(E), T ( p), := + , T ( p) = . , :

T = inf{0 Orth(E) : T ( p)} == inf{ Orth(E)+ : T ( p)} == inf{ Orth(E)+ : T ( p)} = T .

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74 1.

, Orth(E), - , , T = T , , d-. (Tn) Z (p), (n) R - Orth(E) limn n = 0 Tn Tm k m,n k. |Tnx Tmx| kp(x) (Tnx) r- E. , Tx := r-limn Tnx (x X). T Tn k n k, , T Z (p) r-lim T Tn = 0. , Z (p) br- 1.6.8 (2). Z (p) . 1.6.5.

1.7.

1.7.1. (1) XIX XX . - - , . , . [21, 169], . 1930- - [343]. 1960- , , . , . . - [233]. .

(2) , , - . . [220], . . . - . - [359], .-. [480] . . [220]. ,., , [55, 89, 96, 168, 219, 245, 261, 388].

(3) XIX . . - XX . . . - [291]. - . . [2], . . , . . [21], . - [14], . [20], . [374], . [184], . . - [220], . [576], . [349], . [389], . [372]. - - . . - [481]. , ,. [93].

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1.7. 75

(4) 1.1.9 (2) . [314](. [56] [184]). - [184, 220]. .

. n- -, n + 2 , , .

- [378], - (., , . , . - . [56]). - (. [184]).

. n- - , , , -. n+ 1 , .

(5) G P(X) X, X G G G . (X,G ) , G - . , - . . [229]. - , , . . [446]. . , (. [345347]).

(6) 1.1.3 (1, 2) H : M H(M) - , . . : (1) M H(M),(2) L M H(L) H(M), (3) H(H(M)) = H(M). - H : P(X) P(X), (1)(3), () X. -, , H (A B) = H (A) H (B), . :H (A B) = H (A H (B)) = H (H (A) H (B)). - : G H : P(X) P(X)

H (M) :=

{G G : M G}.

, H G := {G X :H (G) = G} (. [229]).

1.7.2. (1) - . - 1.2.1 1.2.1 (4)

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76 1.

. . (. [1]). 1.2.1 : , , , , . . - . . 3.

(2) , . . - 1960- . - - , (. T. - [527]) - (. . - [194], . . [225], . . [212]). . . - . . [79], . . [315], . . . . [444], . . [125], .-. . [204],.-. . [271], . . [212]. . - , .

1.7.3. (1) XIX XX . - . [359]. . . - [220]. , -, .-. [480], - . . - [4, 78, 190, 249, 311, 365, 388].

(2) - , . , . . -, . . , . , . , . . - . . [82] K- , . [8386]. , 1.3, - , . . [220].

(3) 1970- . . . [224] . . [154]. . . . . [1].

1.7.4. (1) ( . [386]). 1.4.13 (1) 1935 . . . - , .

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1.7. 77

. F = R (. [98, 168]) [102]. 1.4.15 . . [85]. - - . . [179]. - .

(2) ( 1.4.13 (2)) . . [292] .-. [570]. , 1.4.10, . . [394]. (- ). . , . [312], . [482].

(3) . . . . [1], - . . [154], . . [224], . . [233]. 1.4.161.4.19 , . [445] . . [448]. . [30, 130, 258, 132].

(4) . , - , , . [237, 391, 395]. [391].

X Y . A X Y , A (x) x X

inf{y : y A (x), x 1} > 0.

, k ,

A (x) := sup{y : y A (x)} kx (x X),

A . f : U Y , U X x0 U .

A f x0 Df(x0) := A ,

limxx0h0

1h inf{f(x+ h) f(x) y : y A (x)} = 0.

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78 1.

f(x0)(h, y) : X Y R

f(x0)(h, y) := inf>0

supx+thx0xx0

{1tf(x+ th) f(x), y

}.

f x0, f(x0) -. 1.5.8 (4) A , s(A ) =f(x0). , Df(x0) = A . - .

. , f : U Y x0 x0. V f(x0) g : V X, g f(x) = x x X, x x0 < .

(5) . [319],. . - [191], . [514] . [27, 585].

1.7.5. (1) - . . - (. [169, 241]). . .

(2) , - , . . - . . , . . . (. [21, 169, 389]).

(3) - . . [179].

1.7.6 (1) , -, . . 1936 . [84]. . - [438] espaces pseudodistancies, . . , . - , . [83, 90, 417, 428, 544]. , ., , [1, 125, 204].

(2) , .. [83] (. 1.6.1 (4)). . 1.6.1 (4) - (. [125, 126]).

(3) - . . [125, 126]. ( -

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1.7. 79

) . [324, 325]. , - 1.6.7 (1, 3), [95].

(4) , . [30, 90, 466]. [406]. . [327], - . - . -, - (., , [125, 150, 160, 243, 288, 349]). - , - . n- 1.6.9 1.6.11 . . [215, 518].

(5) 1.6.5 . . [126] , E . [125] . E . ., . ., . . [95]. (, X = E) . . . . [29].

(6) K- . . , . [295]. , , K- - . 1.6.6 (1), , , , [125]. 1.6.6 (2) bo- - . [125, 147]. X = oX 1.6.6 (3) . . - (. [129]).

1 [30, 55, 98, 99, 125, 169,179, 190, 202, 233, 256, 258, 262, 277, 295, 361, 388, 406, 415, 589].

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2

-, . - -. , - , . - , , - . .

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

, - , . . , . . - , , - , , , .

2.1.

, , K-- -. . , -

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2.1. 81

. . .

2.1.1. . X , E .

(1) P : X E -:

P (x) P (y) P (x y),|P (x) P (y)| P (x y) P (y x),

P (x) P (x) (x, y X).

, P P (x) = P ((xy)+y) P (x y) + P (y). , P (x) P (y) P (x y). P (x) P (y) P (y) P (x) P (y x). , y = 0, P (x) P (x).

X Y . P :X E , x1, x2 X x1 x2 P (x1) P (x2). , - . T T (X+) Y +, Z+ := {z Z : z 0} - Z. X Y , , L+(X,Y ).

(2) P - X K- E , P ,. . P L+(X,Y ).

P T P , x X+ Tx P (x) 0. , T L+(X,E). , P L+(X,E) x1 x2, 1.4.14 (2)

P (x1) = sup{Tx1 : T P} sup{Tx2 : T P} = P (x2),

. 2.1.2. E -

A. l(A, E) () - A E. , l(A, E), EA, f : A E, - {f() : A} E. l(A, E) , -

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82 2.

:

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

E K-, l(A, E) E-

x := sup{|x()| : A} (x l(A, E)). , l(A, E) Orth(E), , Orth(E) f l(A, E) f := f .

(1) E K-, l(A, E) K-, - (f) l(A, E) :

(sup

f)() := sup{f() : },

( inf

f)() := inf{f() : }.

, l(A, E) . . ,

f = || f Orth(E) f l(A, E). bo-- l(A, E) 1.6.5.

A,E l(A, E) E,

A,E : f sup{f() : A} (f l(A, E)),

( A E). , K- , A,E A. n -, A n. n .

(2) . - .

. 2.1.3. A , -

X K- E. (. 1.5.3), A , x X - {x : A} () . Ax -, A x E, . . Ax : x. A , Ax l(A, E) x X. , A : X l(A, E), - : A : x Ax. A

PA : x sup{x : A} (x X).

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PA . PA 1.5.7 - cop(A) A. - .

P , P = cop(A), P = A A.

1.4.14 (2) P = cop(P ). , P : X E A := P . - , , , .

2.1.4. A := A,E E l(A, E), e E e ( A), (Ae)() = e A.

(1)

A,E A,E = IE , A,E A,E(f) f (f l(A, E)),

IE E . .

(2) F K- P : E F .

(P A,E) ={T L+(l(A, E), F ) : T A P

}.

P A,E , 2.1.1 (2) (P A,E) . , T (P A,E) y := Ax, (1)

T Ax = Ty P A(y) = (P A)Ax = Px,

T A P ., , T : l(A, E) F

T A,E P . f l(A, E)

Tf (T A)(A(f)) P A(f),

. . T (P A,E), . (3)

:

A,E ={ L+ (l(A, E), E) : A,E = IE

}.

, (2), P IE .

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84 2.

(4) A :

cop(A) = A A. , 1.4.14 (4) (3):

cop(A) = PA = (A A) = A A.

2.1.5. E , F K-. - S1, . . . , Sn L(E,F ) (S1, . . . , Sn) (e1, . . . , en) S1e1 + . . .+ Snen, En F .

(1) P : E F . - e1, . . . , en E Q(e1, . . . , en) := P (e1. . .en). Q : En F , :

Q =

{(S1, . . . , Sn) : S1, . . . , Sn L+(E,F ),

nk=1

Sk P}.

l(n,E) En. S L+(En, F ) - S = (S1, . . . , Sn), S1, . . . , Sn L+(E,F ). S n,E P, S1 + . . .+ Sn P . , K- E - 2.1.4 (2). 2.1.4 (2) E, - .

(2) S : E F , - Q : En F

Q(e1, . . . , en) := S(e1 . . . en) (e1, . . . , en E).

Q =

{(S1, . . . , Sn) : S1, . . . , Sn L+(E,F ),

nk=1

Sk = S

}.

(1), S = {S}. -

(. 2). K- E. E

: E E, 0 e e e E+, . . - 0 IE . E M(E), M(E) = [0, IE ] - Lr(E). .

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(3) - n :

n =

{(1, . . . , n) : 1, . . . , n M(E),

nk=1

k = IE

}.

(2) T = IE . E E: | | : e |e|,

()+ : e e+ () : e e.(4) :

T (| |) = [T, T ], T (()+) = [0, T ], T (()) = [T, 0],(| |) = [IE , IE ], (()+) = [0, IE ], (()) = [IE , 0].

(2) (3) . ( .)

2.1.6. - .

(1) . P1 : X E P2 : E F .

(P2 P1) ={T P1 : T L+(l(P1, E), F ), T P1 P2

}.

P1 = cop(A1) P2 = cop(A2),

(P2 P1) ={T A1 : T L+(l(A1, E), F ),

( A2) T A1 = A2} . 2.1.3, P2 P1 = P2 A1 A1. -

2.1.4 (2) 1.4.14 (4),

(P2 P1) = (P2 A1 A1) = (P2 A1) A1 ==

{T L+(l(A1, E), F ) : T A1 P2

} A1 == {T A1 : T 0, ( A2)T A1 = A2} ,

. (2) . - K--

E (. . P(E))

(P2 P1) =

TP2((T P1) + (T d P1)),

d := IE , . (1)

(P2 P1) ={S P1 : S L+(l(P1, E), F ), S P1 P2

}.

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86 2.

E0 := (E), l(P1, E0) K- l(P1, E). . , , P1 =P1 . , BP1 P2 B L+(l(P1, E), F ) T := B P1 .

T = B P1 = B P1 ,T d = B dP1 .

, (1),

S P1 (T P1),S d P1 (T d P1)

, , S P1 = (S + S d) P1. ,

S P1 (T P1) + (T d P1). , . -.

(3) P1 : X E , P2 : E F - ,

(P2 P1) =

TP2(T P1).

(2) = IE . 2.1.7.

2.1.6 (1).

(1)

P1, . . . , Pn : X E

(P1 . . . Pn) =

1,...,nM(E)1+...+n=IE

(1 P1 + . . .+ n Pn).

P : X En :

P (x) := (P1(x), . . . , Pn(x)) (x X).

, P n P = P1 . . .Pn. ,

P = P1 . . . Pn := {(T1, . . . , Tn) : Tk Pk (k := 1, . . . , n)}.

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2.1.6 (2) 2.1.5 (3)

(P1 . . . Pn) =

1,...,nM(E)1+...+n=IE

((1 P1) + . . .+ (n Pn)).

1.4.14 (5). (2) E , F K-.

P1, . . . , Pn : X E - S : E F

(S(P1 . . . Pn)) =

S1,...,SnL+(E,F )S1+...+Sn=S

((S1 P1) + . . .+ (Sn Pn)).

P , (1), Q 2.1.5 (2). Q P = S(P1 . . . Pn). 2.1.6 (2) 2.1.5 (2).

Z (E) E

Z (E) := {S Lr(E) : (n N)|S| nIE}.

, Z (E) (. 2).

(3) T A E. f l(A, E) Tf = Tf , . . T l(A, E) E, Z (E).

E. f l(A, E)

A(f) Tf A(f) = A(f). , d := IE d T = 0., T = T . , T d = 0. T =T+Td = T . , T , . . := t11+. . .+tnn, 1, . . . , n t1, . . . , tn . ,

Tf = T

(n

k=1

tkkf

)=

nk=1

tkkTf = Tf.

> 0 , | | IE . , - T ,

|T (f) Tf | = |T ( )f + T (f) Tf | T (|( )f |) + | ||Tf | T (|f |) + T (|f |) = 2T (|f |).

.

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88 2.

2.1.8. X , F - K-. T : X F -, [0, T ] = [0, IF ] T . [0, T ] := {S L(X,F ) : 0 S T} [0, IF ] := { L(F ) : 0 IF } L(X,F ) L(F ) . - -, F Orth(F ). , , - , .1.6 (4). T : X F -, x, y E

Tx Ty = inf{Tu : u X, u x, u y}.

, :

Tx Ty = sup{Tu : u X, u x, u y} (x, y X),|Tx| = inf{Tu : u X, u x, u x} (x X),(Tx)+ = inf{Tu : u X, u x, u 0} (x X).

. - . X , - . , .

X+ X - , X = X+ X+. X+ , X , . . x, y X u X, u x u y.

(1) X T X F . F = {0}, X+ .

X := X+ X+ x0 X \ X. f X , ker(f) X f(x0) = 1. f e x f(x)e(x X). , T+fTx0 [0, T ], Tx0+Tx0 = Tx0 [0, IE ]. Tx0 = 0. , , X = X T = 0. F = {0} e F \{0}, T +fe [0, T ] , , f = 0.

(2) T : X F . X+ , T .

(1). T x / X+ X+, X u, u x, {Tu : u x} |Tx| .

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2.1.9. (1) . X , E K--. T : X E - , S : X E, 0 S T , Orth(E) , 0 IE S = T . , T , T .

P1, P2 : X2 E, P1(x1, x2) := T (x1 x2) (x1, x2 X),P2(x1, x2) := Tx1 Tx2 (x1, x2 X).

, T , P1 = P2. 1.4.14 (2) P1 = P2. 2.1.5 (2)

P1 = {(x1, x2) T1x1 + T2x2 : T1, T2 L+(X,E), T1 + T2 = T}.

, 2.1.7 (1)

P2 = {(x1, x2) 1Tx1 + 2Tx2 : 1, 2 M(E), 1 + 2 = IE}.

. .

(2) X , F K-. T : X F , T .

T , . T = 0, X+ 2.1.8 (2).

: T , S : X F 0 S T . x ker(T ) u X, u x, u x. Sx Su Tu, u |Sx| |Tx| = 0. , ker(T ) ker(S), 0 : F0 F , F0 := T (X), , S = T . , 0 . f = Tx x X, 0(f) Tu u x, , 0(f) inf{Tu : u X+, u x} = f+. 1.4.15 (2)0 : F F , (f) f+ (f F ). 0 IF . ,S = T , T = 0 T .

: , [0, T ] = [0, IF ] T . p : X F ,

p(x) := inf{Tu : u X+, u x} (x X).

, p(x) = (Tx)+ x X. - p(x) (Tx)+, . u X 1.4.14 (1)

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90 2.

S L(X,F ), Su = p(u) Sx p(x) (x X). x 0 Sx p(x) = 0, x 0 Sx p(x) = Tx., 0 S T , S = T Orth(F )+. S,

p(u) = Su = (Tu) ((Tu)+) (Tu)+, .

2.1.10. . X , E K-. ,, X0 X T0 L+(X0, E) X0. T T0 X. : T0 -, T X, .

, . X = {x0 + tx1 : x0 X0, t R} x1 X \X0.

Tx1 := inf{T0x0 : x0 X0, x1 x0} T0 := T X0. , T ( X0). T [0, T ], T X0 =T X0 x0 X0 M(E) ,. . T x0 = Tx0.

T x1 = inf{T x0 : x0 X0, x1 x0} = Tx1;(T T )x1 = inf{(T T )x0 : x0 X0, x1 x0} = (IE ) Tx1,

T = T . X =

t Xt, (Xt) ( )

, X0. , , - Tt : Xt E Ts Tt Xs, Xs Xt. T T0 X, Tx := Ttx (x Xt). P, Pt : X E -

P (x) := inf{Tx : x X, 0 x, x x},Pt(x) := inf{Tx : x Xt, 0 x, x x}.

(1) P Pt , ,

P = [0, T ], Pt = [0, Tt].

- T (Ttx)+ = Pt(x) x t, x Xt. , (Tx)+ = (Ttx)+ = Pt(x) P (x) (Tx)+., (Tx)+ = P (x) x X. , [0, IE ]T = (x (Tx)+) = P = [0, T ], . . T . T X0 = T0.

2.1.11. 2.1.4, 2.1.6 2.1.7 , A,E . -

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. - E := R.

l(A) l(A,R) , , - A A,R. P(A) A. , , X# := L(X,R) ba(A) - : P(A) R. , (A) 0 (A A) (A) = 1. , l(A), - l(A), ba(A). I : l(A) R, . .

I(f) :=A

f() d() (f l(A)).

.(1) A -

A.(2) A X#, cop(A)

, A ,

(x) =A

x | d() (x X).

, x | := (x) ( X#, x X). , A . -

cA A C(A) - A. rca(A) A. C(A) rca(A) . - I . .

(3) cA - A.

(4) A (. . (X#,X)-) X. X# cop(A) -, A ,

(x) =A

x | d() (x X).

- K-. , - 3.

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, (3) (4). (1) (2) 2.4.4.

2.1.12. , - .

(1) ba(A,A , E), A A. St(A,A ) : A R = nk=1 akAk , A A, A1, . . . , An A , a1, . . . , an R. I :St(A,A ) E,

I

( nk=1

akAk

):=

nk=1

ak(Ak).

, I , -

|I(f)| f||(A) (f St(A,A )), f := supA |f(A)|. St(A,A ) - l(A,A ) . , I - ( ) l(A,A ) .

(2) , A - A -. I(f) - f C(A). , I 0 , 0.

E- . - : A E , A A (A) = inf{(U) : U A, U Op(A)}, Op(A) - A. , A A , . , : A E (), - + (). rca(A, E) qca(A, E) E- - . , rca(A, E) qca(A, E) ca(A,A , E). , () - ca(A,A , E) () . . , qca(A, E) rca(A, E) K-.

(3) , , A E K-. - C(A)E l(A, E),

nk=1 k ek,

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ek E k C(A), n

k=1 k()ek ( A). - Cr(A, E) C(A, E) br- bo- C(A) E l(A, E), . .Cr(A, E) := r(C(A) E) C(A, E) := rd(C(A) E) ( 1.6.6). ,C(A, E) bo- C(A)E( l(A, E) ) (. 1.6.6 (3)).

(4) qca(A, Lr(E,F )), I Lr(Cr(A, E), F ) , I( e) = (

A

d)e C(A) e E. I(f) =

A

f d. T

Lr(Cr(A, E), F )

Tf =A

f() d() (f Cr(A, E)),

qca(A, Lr(E,F )).(5) L(C(A, E), F ) -

T : C(A, E) F , - : () P(E) f C(A, E) Tf =

T (f). qca(A, Ln(E,F )), -

I Lr(C(A, E), F ) , I( e) = (A

d)e

C(A) e E. I(f) =A

f d.

T L(C(A, E), F )

Tf =A

f() d() (f C(A, E)),

qca(A, Ln(E,F )). . 3.2.1.13. rA :=

rA,E

A :=

A,E

A,E Cr(A, E) C(A, E) .(1) P : E F

(P rA) ={I() : qca(A, Lr(E,F ))+, (A) P

}.

r Cr(A, E) l(A, E), 1.4.14 (4)

(P rA) = (P A r) = (P A) r. 2.1.4 (2) 2.1.12 (4).

(2) o- P : E F

(P A) = {I() : qca(A, Ln(E,F )), (A) P} .

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C(A, E) l(A, E). 1.4.14 (4)

(P A) = (P A ) = (P A) . 2.1.4 (2) 2.1.12 (5) .

(3) :

A,E ={I() : qca (A,Orth(E))+, (A) = IE

}.

(2) P := IE , qca(A, Ln(E,E))+, (A) = IE (A) Orth(E) A A .

(4) K- E (,)-, A,E =

{I() : rca(A,Orth(E))+, (A) = IE

}.

2.1.14. . p : X A E , x p(x, ) (x X) A, p(x, ) ( A) r- x X.

q(x) := sup{p(x, ) : A}. q : X E

q ={

(A

p(, ) d())

: qca(A,Orth(E))+, (A) = IE}.