ФАН - конспект лекций

ε

3 9

$ ( $

(X, ρ) % & % M X ε > 0 % &

% A X ε%& M x M a A : ρ(x, a) < ε

√

X = R2 = M, A = {(m, n)|m, n Z} A	2	%
	2
8 % M & % (X, ρ) % ε > 0			&
ε% M
M & &			= {a1, . . .
/ ε = 1 M & & % M & A1

. . . , an}. x0 M r0 = max{ρ(x0, a1), . . . , ρ(x0, an)} x M ρ(x, x0 ) ρ(x0, ak)+ρ(x, ak )

r0 + 1 < r0 + 2 M B(x0, r0 + 2).

/Rn & % M &

! * ε > 0 M √εn & ε% "

& & ' ε%

> , #

S = {x = (x1, . . . , xk , . . .)| ∞ |xk | = 1} l2 >& & S & &

&

k=1

√

√

n

n

m

2

) =

2 (m = n)

/

&

ε0

< 2 &

x = (0, . . . , 0, 1 , 0, . . .) ρ(x

, x

n−

ε0% S Aε0

= {a1, a2, . . . , an}

n0, m0, k : m0

= n0 ρ(xm0 , ak) < ε0, ρ(xn0 , ak ) < ε0

√

*

√

& &

ρ(xm0 , xn0 ) ρ(xm0 , ak) + ρ(xn0 , ak ) < 2ε0 = 2

ρ(xm0 , xn0 ) = 2

! C , / 2A " 3 8 & %

&

ε > 0

x1

X x2 X : ρ(x1, x2)

ε &

X %

x X ρ(x, x1)

< ε {x1} % & ε% x3 X

: ρ(x3, x2) ε ρ(x2, x3)

ε &

" {x1, x2} % & ε% 6 .

* % " & ε% & &

% xn : ρ(xk , xn) ε, k = n / (X, ρ) {xn} @ %

{xnk } ρ(xnk , xnk ) ε & & @ 9 &

	X & {xn} {εn} : nlim εn = 0, εn > 0			ε1% X & )
	→∞				m1			xn
	" B(ak , ε1) * & &
& Aε1 = {a1, . . . , am1 } Aε1 & (X, ρ) & X					k=1 B(ak , ε1)
					%	{				}	/ %
			xn								m2
* * n1 ! ε2% X Aε2				=	{b1, . . . , bm2		}		X		k=1 B(ak , ε2)
" & & %			{ }							' "
ε1
* & * %		" %

# " & % {xnk } '# * % ' {xn} '

& m > l xnm , xnl " εl > ' & {xnk } @ (X, ρ) %

A 4 4 -4

! D (-D ( 2 " C[a;b]3 8 Φ C[a;b] %

.	ϕ Φ |ϕ(x)| M
Φ & M R : x [a; b],
Φ * ε > 0	δε > 0 : x1, x2 [a; b] : |x1 − x2| < δε ϕ

Φ |ϕ(x1) − ϕ(x2)| < ε

+ Φ M @ Φ & ε >

> 0 ε − Φ / ε3 %

|ϕi(x)| < ki		x [a; b] k =
max ϕ(x)			−	ϕ (x) <
ρ(ϕ, ϕk ) = x		[a;b] |		k	|

Φ				{ϕ1	, . . . , ϕk0	} 9 i = 1, k0		ki :
		ε
max ki +		3	|ϕ(x) |ϕ(x) − ϕk (x)| + |ϕk (x)|, ϕk &

|ϕi(x)| < k i ϕ Φ

Φ&

/( ϕ1, . . . , ϕk [a; b] ε > 0 δε,i > 0 :

[a; b] : x

ε

= min

ϕ(x )

x

, x

|ϕi(x1) − ϕi(x2)| <

i

= 1, k0

δ

δ

1

2

| 1

− x2| < δε,i

3

ε

ε,i

|

1

−

i=1,k0

− ϕ(x2 )| |ϕ(x1 ) − ϕk (x1)| + |ϕk (x1) − ϕk (x2)| + |ϕk (x2) − ϕ(x2)| < 3ε + 3ε + 3ε = ε 2 Φ

*

+ Φ & 3 & Φ %

/ M @ & ε% ε > 0 ε > 0 δε

|x(t + s) − x(t)|p dµ

6 ) [a; b] & xi δε x0 = a < x1 < . . . < xn = b >

[−M ; M ] y !5( ) " 5ε

& yi. y0 = −M < y1 < . . . < yn =

−

ε

[a; b] ×[−M ; %

= M Y [a; b] [ M ; M ]

M ]

* x " δε * y " 5 6 * @ * Φ

xi ' & yi [xi, xi+1] ' * >& &

*

&

ε% ' Φ x

[x , x

] &

Φ

Φ : ϕ(x )

! & Φ

ϕ

ψ

−

ε

k k+1

3ε

|

k

− ψ(xk ) <

5 |ϕ(xk+1 ) − ψ(xk+1 )| |ψ(xk )

3−ε ϕ(xk )| + |ϕ(xk ) − ϕ(xk+1 )| + |ϕ(xk+1 ) − ψ(xk+1)| <

5

ε

|ε

k3ε

−

ψ(x)

|

<

5 6

|

−

| |

−

k |

|

k

−

k

|

+

ϕ & % *

ϕ(x )

ϕ(x)

ψ(x)

ϕ(x)

ϕ(x )

+ ϕ(x )

ψ(x )

+ ψ(xk )

ψ(x)

< 5 +

5 +

5

= ε 6

[xk ; xk+1]

ψk

Φ :

ρ(ψk , ϕ) < ε > '

|

−

|

ϕ

Φ

ρ(ψ0, ϕ) < ε Φ

[a; b]

ϕ0 Φ :

&

ε% Φ

! 2A " lp3 M lp

M & lp K : x M

∞ |xk |p < K

=1

∞

p

k

k=

|xk | < ε x M

ε > 0 N N :

N +1

! 2A " Lp[a,b]3 M Lp[a,b]

x

M

%

ε >

0

δ

>

0 :

s

(0, δ)

M % & K : x M

b

|x(t)|p dµ < K

a

M

a

1

< ε

p

b

"

" 2 3

) + % K + *" K = R K = C X =

+ . K. ” + ” : X ×

×X → X ” · ” : K × X → X '# '# .

X % ' .

x + (y + z) = (x + y) + z, x, y, z X

x + y = y + x, x, y X

, O X : x X x + O = x

? x X − x X : x + (−x) = O

(αβ)x = α(βx), α, β K, x X

1 · x = x x X

, (α + β)x = αx + βx, α, β K, x X ? α(x + y) = αx + αy, α K, x, y X

		X = R &
	X = Rn
		X = Cn ' & #
		X = C[a,b] (f + g)(x) = f (x) + g(x);					(λf )(x) = λf (x) !
& @ * x & R * %
		X = lp, p 1		x + y = (x1 + y1, x2 + y2, . . .); λx = (λx1 , λy1, . . .) Z % ZZ
* R
		X = Lp[a,b] ' ? & &
	1	X = c % % # &						! & @
Lp	@ *
			x + y = x + y, λx = λx

* x @%

% * > :

8 X = c0 % % # D & % * >

: ! & : & & & % &

9 X = l∞ % % & * 5 :

: X = F (T ) % % @ * T 5 ?

X % * % + 5 x1, . . . , xn +

a1x1 + . . . + anxn = 0 a1 = . . . = an

X % 5 x1, . . . , xn, . . . +

' & *

& * ' * & * + & *

X % ' & (dim X = ∞) & *

		C[a,b]			% & %
		xα		=	t−a	+ 1, a t α
	{		}		a−α
					0, α < t b

xα1 , . . . , xαn 8 & & α1 < . . . < αn λ1xα1 + . . . + λnxαn = 0 t [a, b] /

t = αn−1 xαi		= 0, i = 1, n − 1 λn = 0 & & & λi = 0
	lp,	p 1	xn = (0, . . . , 0,	1 , 0, . . .) λ1xα1	+ . . . + λnxαn =0 %
* αi λi				n−		lp % &
				D1RD > ' &
	Lp[a,b] % & % 5

#

< X ' X @ + || · || : X → R

.

||x|| = 0 x = 0

||αx|| = |α| · ||x|| α K x X , ||x + y|| ||x|| + ||y|| x, y X

3 % & % * ρ(x, y) = ||x −y|| ! *

% & , . ρ(x, z) + ρ(z, y) = ||x − z||+ ||y − z|| ||x − z + z − y|| = ||x − y|| = ρ(x, y)

/ % R, Rn, C, C[a,b], lp, Lp(X) % D

3 ρ(x, y) = ||x −y||

%

' % ( : %

, C1[a,b] %

X % || · ||1 || · ||2 ' ! X C1 C2 = Const, C1, C2 >

>0 : C1||x||1 ||x||2 C2||x||1 x X

|| · ||1 || · ||2 '# %

% % / Rn

- ( 5- -

X % L X, L X *

K

R % R2

L % % X / X " '# . x y

x − y L > X '# (

x & x / K

X

'# .

%

x + y = x + y, λx = λx

' $

X

L

& '

X/L

f (x)

0 X/L = L (X )/N = L (X )

X = L (X ), L = N =

L (X ) f (x) . .

p

{

p

|

}

p

p

1 " /, (

1 ( ( " 4

X1, X2 % * + % + K > A : X1 → X2

.

x, y X1	A(x + y) = Ax + Ay
x X,	λ K A(λx) = λAx
< * * A : X1 → K K = R C) $
		Ax = x * & * *
		Ax = 0 * * *
		F : X1 → K X = l2, K = R F (x) = x1 + x2 x = (x1, x2, . . . ) * * @

A : Rn → Rn,	A =
*

a11	a12 . . .	a1n
a21	a22 . . .	a2n

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

an1 an2 . . . ann

(x) A * *

1 " /, (

< * * A : X1 → X2 % 5 > 0 : ||Ax|| 5 ||x||X1 x

X1

1 < * * A : X1 → X2 % ' & %

&

< * * @ F : X → K % C 0 : |f (x)| C ||x|| x

X

2 < * * @ f : X → K % ' & %

X & % K

* 1 1

1 1 & % M

X1.

A

x M. x → Ax. ρ(Ax, 0) = ||Ax|| C ||x|| . /

& M, ||x|| < +∞ x M = {Ax|x M } &

1

1.

& % M

X

M

X

&

x

A

1

2

M. x → Ax. / & M, M ρX1 (x, 0) = ||x|| < +∞,

ρX2 (Ax, 0) = ||Ax|| < +∞,

C = sup

||Ax||

0 ||Ax|| C ||x||

x X1

||x||

x=0

* 2 " *

A : C[a;b]

→

C

[a;b]

2

x(t), A

* *

||Ax|| =

max

t2x(t)

, Ax(t) = t

t [a;b]

|

|

max

2

max

x(t) =

t [a;b] |t

| · t [a;b]

|

|

max{a, b} · ||x||

b

F : C[a;b] → R,

b

F (x) =

x(t)dt * * @

a

b

A :

C[a;b]

b

Ax(t)

K(t, s)x(s)ds K(t, s)

[a; b] × [a; b]

→

C[a;b],

=

→

a

|

|

|

| | 1|

| 3| || ||

2

2

,

1

1

3

1 2

− a)

1

|

3

| t [a;b] a

K(t, s)x(s)ds

|

t [a;b] a

|K(t, s)x(s)|ds = M ||x|| (b

M =

t,s

|

max

max

max K(t, s)

f : l

R

f (x) = x

+ x ,

x = (x , x , . . . ).

f (x) = x

+ x

x

+ x

2

x

A : C[a;b]

→ C

[a;b] .

||x||X1 = t [a;b] | |

=

||

||

||

||

c

||

||

(Ax)(t) = Const

max x(t)

x

Ax

x

ρ(x

max x (t) = max

1 sin nt

0.

ρ(Ax , A(0)) =

= max

cos nt

= 1

0

||xn||

|

1

→

||

Ax

|

|

→

n, 0) =

= t [a;b] |

n

|

t [a;b] |n

n

n

n||

t [a;b]

! 1 A : X1 → X2 * * X1, X2 5 '#

.

A & x0 = 0 X1

A X1

, A ε > 0

δ > 0

x1, x2 X : ||x1 − x2|| < δ ||Ax1 − Ax2|| < ε

? A &

3) 2) 1)

& ! &

4)

3) 1)

4)

ε

4) 3) ||Ax1 − Ax2|| = ||A(x1 − x2)|| = c||x1 − x2||,

δ =

c !

1)

4) A x0 = 0

ε > 0

δε > 0

x

X : x

x0

||

< δε

Ax

Ax0 <

|| −

||

−

δ||x

< ε x = 0

Ax = A(0) = A(x

−

x) = Ax

−

Ax = 0 ! ε = 1,

δ

: x

< δ

1

Ax

1 ) x

=

1

2||x||

||x || =

0

1

||

||

||

||

δ x

=

δ1

δ1x

δ1

< 1;

||Ax|| <

2

||x|| .

2||1x||

2

< δ1

||Ax || = A 2||x||

= 2||x|| ||A(x)||

δ1

1

#

( "

A : X1 → X2 * * & * 1 A + & ' ||A||

' " ' & ' .

||Ax|| ||A|| ||x||

ε > 0 xε X1 : ||Ax|| > (||A|| − ε) ||xε||

! 1 A : X1 → X2 & * ||A|| & * '# @ .||A|| = sup ||Ax||

||x|| 1

||A|| = sup ||Ax||

x=0 ||x||

, ||A|| = sup ||Ax||

||x||=1

||x||

1 A & c > 0 : ||Ax|| c ||x||

||Ax|| c

x : ||x|| 1 {||Ax||}

&

sup

||Ax|| & ||A|| 3 & ε > 0

xε

:

> ( A

ε)

||x|| 1

=

xε

= 1

X

1

||

Ax

|| −

x

x

,

x

||xε ||

||

||

||

ε||

ε

||

ε||

Ax

=

||Axε||

=

1

Ax

> ( Ax

ε)

x

||xε ||

||

ε||

||xε||

||

ε||

||

|| −

|| ||

||Ax||

ε > 0

xε

: ||xε|| 1 ||Axε|| > ||A|| − ε ' & * * ||A|| =

sup

||x|| 1

) & +,

@ + .

||Ax||

x

X

,

x

= 0

=

Ax

x

X

,

x

= 1

}

4Ax

x

A : X

X

# ||x||

|

1

|| || $

{||

|| |

1

||

||

=

X Ax = x,

Ax

=

x

A

=

sup

Ax

||

||

A(

=

||x||

)

||x||

→

||

||

||

||

||

||

||x||=1 ||

||

=

sup ||x|| = 1

||x||=1

A : C

→ C

,

(Ax)(t) =

2

)x(t),

[a; b]

= [0; 1].

||Ax|| = t

2

)x(t)|

[a;b]

[a;b]

(2 − t

[0;1] |(2 − t

max

max

x0(t) = 1

||x|| = 1

||Ax|| = 2

||x0

2 t

[0;1] |x(t)| = 2 ||x||

|| = 2 ||Ax0||

'# . & &

+ " *	# 9 %
* *
1 4$ ( ($ ($ 4$		: X1	→ X2 {An}
{An} % * & n N An
( * & A ||An − A|| → 0
{An}	(	A	x
X1 ||Anx − Ax|| → 0 n → ∞

{An} A

||Anx − Ax|| = ||(An − A)x|| ||An − A|| ||x|| → 0 x X1

8 ( " 4 4

> & & L(X1, X2) % % * & * '#

X1 X2 >

'# .

(A + B)(x) = Ax + Bx x X1 + (λA)(x) = λAx x X1 +

! 8 % L(X1, X2) + + * *

*. A = sup	Ax		X2
	x & %
x=0

! & L % & (A + B) L(X1, X2), A, B L(X1, X2)

(A + B)(x) = Ax + Bx Ax + Bx A x + B x = ( A + B ) x A + B = C

8 & A + B A + B +,

4 & & λA L(X1, X2) / * L(X1, X2)

* *

+A = B Ax = Bx x X * X2 2 L(X1, X2) % *

! & 9 &

A A L

&

% &

Ax

C

A

x

1 .

A

= 0 0 = sup

Ax

Ax

= 0

x

X

, x = 0

x

x=0

1

>& & x = 0 Ax = 0 A(0) = 0 Ax = 0 x X1 4 = 0 % * /

	' &
	2 .		λA		= sup	λAx	= sup	|λ| Ax	λ sup	Ax	= λ A
						x				x
					x=0			x	| |		| |
									x=0

+, L %

, X2% ! & L(X1, X2) % {An} % @ %

L(X1, X2)

ε > 0 Nε N : n, m > Nε An − Am < ε

x X1 (An − Am)(x) An − Am x @ x X1 {An} % @ X2 X2 % +

lim Anx = Ax, x X1 +?

n→∞

& 4 * @ * +? *

A(x + y) = nlim An(x + y) = nlim (Anx + Any) = Ax + Ay ||Anx + Any − Ax − Ay|| ||Anx − Ax|| +
→∞	→∞

+||Any − Ay|| → 0

4 & A(λx) = λAx

2 4 % * * 2 @ * %

&

m0 N : m > m0 Am − Am0 < 1 Am = Am − Am0 + Am0 Am0 + 1

C = max{ A1 , .., Am0 , Am0 + 1}

9 Am C m N +:

8 & 4 % & 9 Am & Amx Am x& +: & & Amx c x +A

9 x = ρ(x, 0) * @ ' @ * * %

% +A * & +? & & Ax c x

4 % & A L(X1, X2)

2 @ ε > 0 Nε : n, m > Nε ||Anx − Amx|| = ||(An − Am)x|| ||An − Am|| · · ||x|| < ε||x||

m → ∞ @ x & & ε > 0 Nε : n > Nε ||Anx −

− Ax|| = ||(An − A)x|| < ε||x|| & & ||An − A|| < ε A = lim An

n→∞

8

X % Rn > & & X & * @ %

M

X = {f |f : X → K(K = R C) f % & * [

X % =

X = L(X, K) 9 X % ( %

8 "

4, / % * . A : X1 → X2, B : X2 → X3

(BA)(x) = B(Ax), x X1 BA : X1 → X3

9 X1, X2, X3% * /4 % * *

! * (BA)(x + y) = B(Ax + Ay) = B(Ax) + B(Ay) = BA(x) + BA(y) 4 & (BA)(λx) =

=λ(BA)(x)

A, B % * & % BA & * ||(BA)x|| =

=||B(Ax)|| ||B|| · ||Ax|| ||B|| · ||A|| · ||x||

4 % * * & * A : X1 → X2 f X2 9 f A X1

+f : X2 → K, f A : X1 → K A : X2 → X1 @ A (f ) = f A

4

4 % * * & * A * * & * &

A A

! & A * * & A (αf1 + βf2) = αA (f1) + βA (f2)

A (αf1 + βf2)(x) = ((αf1 + βf2)A)(x) = (αf1 + βf2)(Ax) = αf1(Ax) + βf2(Ax) = α(f1A)(x) + β(f2A)(x) =

=αA (f1)(x) + βA (f2)(x) = (αA (f1) + βA (f2))(x) A % * *

8 A f

A f = f A A f

! 2 " 4 3

(A + B) = A + B(αA) + (αB) = α(A + B)

,(AB) = B A

% / '

! , A : X1 → X2 B : X2 → X3

A : X2 → X1 B : X3 → X2 B BA : X1 → X3B(BA) : X3 → X1 f X3 (BA) (f ) = f (BA) (A B )(f ) df A (B (f ))

/ x X3 f (BA)(x) = f ((BA)x) = f (B(Ax)) = (f B)(Ax) (f B)(A)(x) = (f B)(Ax)

9 /, ($ (

9 "

! 26 4-" , 3 X1 % % X2 % % M L(X1, X2) + M % % * & * '# X1 X2 & x X1 Cx : ||Ax||Cx A M ' C : ||A|| C A M

9 F ||A|| = sup ||Ax|| ! & ||Ax||

||x|| 1

& & " % V = {||Ax|| X1| ||x|| 1, A M } &

X2

! & ||Ax|| & % " & &%

"

||Ax||

&

B[x0, r0]

C0

R : x B[x0, r0] A M ||Ax||

1

C0 y =

(x − x0), x B[x0, r0], ||y|| 1 x = r0y + x0

r0

1

A

1

(x − x0) =

1

1

||Ay|| =

||A(x − x0)||

(||Ax|| + ||Ax0 ||)

(C0 + C0) 9 &

r0

r0

r0

r0

Const

& "

&

! V &

" / " B[x0, r0] >& & V & " B(x0, r0) x1

B(x0, r0) A1 M : ||A1x1|| > 1

A1 & ε > 0 δε : ||x − x1|| < δε ||A1x − A1x1|| < ε

1 B(x1 , r1) :

! & B(x1, r1) B(x0, r0) : ||A1x|| > 1 x B(x1, r1) r1 > 0 xr

||A1xr1 ||

1

/

A1

x1

ε0

=

||A1x1

||−1

> 0

δε0 :

||A1x1||−1

2

x X1 : ||x − x1|| < δε0 ||A1x − A1x1||

2

||A1x1||−1

r = δ

ε0

'

x

:

x

x

1||

< r

Ax

Ax

||

<

1

r1

||

r1 −

1 ||

r1 −

1

2

A x

= A x A x

+ A x

< A x A x

+ A x

||A1x1||−1 + 1

|| 1 1||

||

1 1 −

1 r1

1 r1 ||

|| 1 1 −

1

r1 ||

||

1

r1 ||

2

||A1x1||

1

||A1x1||

1

& &

||A1x1|| > 1

& &

B(x1

, r1)

2

2

r0

B(x0, r0) : ||A1x|| > 1 x B(x1, r1) 8 & & r1 <

9 % Ax %

2

& "

B[x1, r1]

6 & & &

B[x2, r2] B[x1, r1] : x B[x2, r2] ||A2x|| > 2

r0

A2 M r2 <

22

& % " B[xn, rn] D 2 {An} : ||An|| > n x B[xn, rn]

9 X1 % " % &

x0 B[xn, rn] n N * . ||Anx0|| > n n N &

& & & V &

( % {An}, An L(X1, X2) A A & A

L(X1, X2)

& {An} & L(X1, X2) {Ax} ||Anx|| = ||Anx−Ax+ Ax|| ||Anx−Ax||+ ||Ax|| <

< 1 + ||Ax|| + n > N0 C = max{||A1x||, . . . , ||AN0 x||, ||Ax|| + 1}

& ||Anx|| < C n N 9 % 9 = % * %

* '& & ' C : ||An|| C n N > ' ||An|| ||An|| · ||x|| C||x||

n → ∞ & ||Ax|| C||x|| A &

9 ; , ( /, ( ! C-6 4

X % p : X → R @

% . p(αx + (1 − α)y) αp(x) + (1 − α)p(y) x, y X, α [0, 1]

X % p : X → R @

p(αx) = αp(x) x X, α R : α > 0

* * @

|| · || % * @ + &

p % * @ p(x + y) p(x) + p(y)

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

L1 L2 L1, L2 % < * * @ F2 : L2 → R *%

@ F1 : L1 → R F1(x) = F2(x) x L1

! 2C-6 4 3 p : L → R % * @ L0 % * % L

' * @ F0 : L0 → R : F0(x) p(x), x L0 F L

& . F (x) p(x), x L

= & & L0 = L z L : z L0

L1 = L(L0, z) = {tz + x0|x0 L0, t R} L1 % L1 L

F % F0 L1 F (tz + x0) := tF (z) + F (x0) = tF (z) + F0(x)

C = F (z) >& & * F % * * @ F (t1z + x0 + t2z + +y0) = F ((t1 +t2)z +(x0 +y0)) = (t1 +t2)F (z)+F (x0 +y0) = t1C +t2C +F (x0)+F (y0) = F (t1z +x0)+F (t2z +y0)B

F (α(tz + x0)) = F (αtz + αx0 ) = αtC + αF (x0) = αF (tz + x0)

3 * ' C & & & C :

F (tz + x0) = tC + F0(x0) p(tz + x0) (1)

t) p(z + xt0 ) (2). t < 0 : C + F0( xt0 ) −p(−z −

−xt0 ) (3).

y , y L0 F0(y ) − F (y ) = F (y − y ) p(y − y ) = p(y − z − (y − z)) p(y + z) + p(−y − z)

−F0(y ) − p(−y + z) −F0(y ) + p(y − z)

> & & C

= sup

{−

F (y )

−

p(

y + z) ; C

=

inf

{−

F (y ) + p(y

−

z)

1

y L0

0

−

}

2

y L0

0

}

& C1 C2 C : C1 C C2

9 F0 L1 & *

L L = {α1x1 + α2x2 + . . . + x0|αi R, x0 L0, xi Li} (4)

& * @ F0 L1 L2 F L

L +? % ΦL0 % * F0

" % L / " '# . & &

F F L L +F L F L 8 % ΦL0

" * & & % ''

9 ] .

F & & & % & % ''

% *

*

> B : L → R +B % K = C p(x +

+ y) p(x) + p(y) 2) p(λx) = |λ|p(x), λ C, x L

9 M %= * & L % C '# %

. * @ # ' &

'# * . |F0(x)| p(x), |F (x)| p(x) +F0, F : L → C

( X % % L % * % f % * * & * @ %

L F X & ||F || = ||f || F %

p(x) = ||f ||·||x|| 9 M %= F : |F (x)| ||f ||·||x||, x X ||F || ||f ||

9 ||F || =	sup	|F (x)|
||f || =	||x|| 1, x X
	sup	|f (x)| / & F (x) = f (x), x L ||F || ||f || ||F || = ||f ||
||x|| 1, x L

( X % % x0 X, x0 = 0 f : X → K : f (x0) = ||x0|| ||f || = 1

L0 = {tx0|t K} f0(tx) = t||x||, t K

f0 % * * & f0(x0) = ||x0|| |f0(tx0)| = |t| · ||x0|| >& & ||f0|| = 1 F : X → K :

||F || = ||f0|| = 1

( A % * * & * A : X1 → X2 ||A || = ||A||6 & ||A || ||A||

x X : x = 0, y = Ax = 0 F : X2 → K : F (y) = ||y|| = ||Ax|| ||F || = 1

||Ax|| = F (y) = |F (y)| = |F (Ax)| = |(A F )x| ||A || · ||F || · ||x|| = ||A || · ||x|| ||A|| ||A || ||A || = ||A||

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