Spectral Theory in the Hilbert Space, Volume 4

TOKYO MATHEMATICAL BOOK SERIES VOLUME IV SPECTRAL THEORY IN THE HILBERT SPACE BY HIDEGORO NAKANO 1953 JAPAN SOCI...

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TOKYO MATHEMATICAL BOOK SERIES

VOLUME

IV

SPECTRAL THEORY IN

THE

HILBERT

SPACE

BY HIDEGORO NAKANO

1953 JAPAN SOCIETY FOR THE PROMOTION OP SCIENCE Ueno Park, Taitoku, Tokyo, Japan

PREFACE

Spectral theory was considered first

symmetric operators spectral theory bounded

on a sequence space.

take a new form*

symmetric

operators,

by D. Hllbert

about bounded

J. von Neumann made this

He generated this theory

defining abstractly

to un-

the Hilbert space.

In this new style spectral theory was discussed by many mathematicians: P. Riesz, M. H. Stone, etc.

On the other hand

as an a Detract ion

near spaces* space.

in semi-ordered li-

a spectral theory was born

of the spectral theory

in the Hilbert

This spectral theory

is constructed

by means of generated,

considering

every spectrum

as

Stieltje*s integral,

On the contrary by the author,

there is

a continuous one.

another type of spectral theory

which is constructed

established

by means of generated Riemann in-

tegral in topological spaces considering every spectrum as a point one*

This new spectral theory is explained

precisely

in

Volume II of this

Series, Modern Spectral Theory, Tokyo (1950), in the name of the second

spectral theory. dpeJttraltheorie,

The auther has remarked in an earlier paper, Bine Proc. Phys.-Math. Soc. Japan, 2)(l94l) 485-511,

that

this spectral theory also is applicable to the Hilbert space, but it is

not well known.

The purpose of this book is to make this new spectral

theory revive in the Hilbert space.

In this text spectral theory is constructed firstly about elements of the Hilbert space, and as its application, we consider the theory of the normal operators.

Furthermore,

*{ctral

by means of this new

ii

spectral theory, we can discuss

not only

the mean ergodic theorem ob-

but the individual ergodic theorem

tained by J. von Neumann,

by J. D. Birkhoff , as a problem in the Hilbert space.

obtained

Readers will

be more interested, if they read this text, comparing Volume II of this

Series,

This book was written

matical Book Series. to publish this book.

during 1950-51 as Volume IV of Tokyo MatheBut financial difficulties made it impossible

Recently

I

have obtained financial help from

Education Ministry of Japan, and it has become possible to publish this book.

I

express my warmest thanks to those who have made effort to

publish this book.

Sapporo,

June 24,

1953

iii

TABLE OF CONTENTS

INTRODUCTION

1

CHAPTER 1.

2

.

3.

4. 56.

7. 8.

9. 1

.

Completion Orthononnal Systems Dimensions of Hilbert Spaces Weak Convergence

11 .

13 . 14.

15 16. 17. 18. 19 . 20. 21.

.

'7

.

J28. 29* 30.

31.

$33. $34*

J35

536.

.20

*

.25 .29

23

*

32

SPECTRAL THEORY

II.

41 44 .49

53 .60

Proper Spaces Integration Integration by Elements.... Relative Spectra Properties of Relative Spectra Properties of Rings. . . . Ring Convergence.

66 72 79 85 .89 ,

94

DILATATOR ANALYSIS

Dilatators

101 108 116

Speotralitiea Spectralization of Dilatators Calculus of Dilatators

123 134 142 147 152 165

D-convergence Measurable Functions General Spectralities Integral Operators Functions of Dilatators

NORMAL OPU31ATORS

IV.

Fundamental Definitions* , Regular Part Properties of Symmetric Operators Specialization of Normal Operators Graphs of Operators Regularity. * Self -ad joint Operators* ..,.* Isometric Operators .**..... ,

|36. 837.

*

, , .

Projection Operators Calculus of Projection Operators Systems of Projection Operators Rings of Projection Operators..,..

CHAPTER

532.

...,.14 1?

35

CHAPTER III. |22 23. 24 . 25. 6.

8 .10

Operators

CHAPTER

12.

HILBKRT SPACES

I.

, . Linear Spaces * . . . Homed Spaces Adjoint Spaces Inner Product Spaces . . . * Adjoint Spaces of Inner Product Spaces*

,

175 181 ,184

189 195 197 .201

206

TABLE OP CONTENTS

iv

Canonical Form Hermitean Operators

5)9-

^40.

210 214

CHAFm 41 42.

$4546.

49.

CHAPTER

53.

55*

22? 253 23/ 242 251 255 257 266

VI.

ERGODIC THEOREMS

firgodic Theorem of founded Linear Operators

50.

54-

220

Isomorphisms of Specialities Unitary Trans format ion of Dilatators ( L i)-spacea Regular 6 -ideals

47. 48.

52 .

UNITARY INVARIANTS

Multiplicity Resolutions Multiplicity Functions rf-ideals of Sets

J4544.

51.

V.

Isomorphisms of Rings Rings of Uniform Multiplicity

t

Ergodic Theorem of Dilatators Translators Ideal Spaces Reduction Theory Measure Preserving Transformations

PROOK OF

APPENDIX.

RdKEREtCES BIBLIOGRAPh

CC

=

,

276 278 279 285 292 294

298

298 't

299

INTRODUCTION We can modify objects for consideration, and we obtain a notion of

Let ft be a space.

spaces .

be a point or an element of

a

mean that

A

$

,

and

Let

ft

R

*

every element

Oi

( X

and

an element 0i(x

)

X

X by

A

of>

* ? X or X

,

X

implies X

8

subset of X

C

ft

,

A

6

and

,

if 8

^

R

,

A

of

x

R

X

R

about

(*)

x

%

of

if

,

R

into the space

x

:

C

j?

,

* *

into the space of sets of * * A

)

^=

{

,/4

A

by

.

and that

,

R

of

C B or 3 o A

&

,

is called

,

,

X A

to mean that

8

or that

a

is

:

x?

we shall denote by y

COO

^

of

A

\.

as

,4

A

Considering every set of

R

R.

A mapping

;

X

R

/4 X (

\

is called a set system.

/U

AA

for some

X

<

*

/! j ,

as an

A For

~

we define the union

X

For a

is satisfied.

M/t J51

*

A set, which

A*

includes

c(x)

we define the complement

(

a

corresponds to

if

,

(

,

For

.

at

be an other space.

set system Ay.

&

.

element, we obtain a space of sets of

of A

R

{

.

correaponds to

With this definition, we have obviously fr" = /I

to

is called the empty set, and

if and only if

/'= lx

Let

to

With this definition, we have obviusly that

for ev8rv 8et

the set which contains

of

R

to mean that

we define A

{

A

X

We shall say that A

.

For a condition

set

by

contains an element *

tne whole space, and denoted by

For two sets

?

which assigns

,

A set, which contains every element

denoted by

R

.

to mean that

does not contains every element

C A

*

does not contain an element

,4

is said to

or

K.

is called a set of

A or A 5

A

We shall say that a set

a set

and

8

a,

is called a mapping of

,

^

A mapping A of

we d fine x 4

and

/4 f

6

)

X

the ima^e of

c^moosed of two elements set

We shall define

a correspondence Oi

,

be a space.

R

object modified by

fiacji *

R

is an element of

For two spaces

into

R.

mean

)

a

2

Tl

and the intersection

=

7T A.

to mean

A*, I

x

X

'

6

/t

A

for all

x

4

/I

K

With this definition we have obviusly

*'or

a set sequence

(v = 1, 2, ...) ve may write

A^

For two sets A and 5, if

/4 :>

Bi then we denote

/4

&' by

/|

~ 8

that

,

is,

X-

=

8

ix

*?g}

^^>)/

:

of

As a method for infinite process we are permitted to make use the following axiom due to Zermelo:

For any space

Choice Axiom.

R

of seta of imago of

A

into the space

ft

we can find a mapping of the space

fl,

such that every set

of

/)

contains the

A * 0.

if

By virtue of Choice Axiom we can prove the following theorem

due

to Zorn:

Maximal Theorem >

Let

C( ** t*a

every finite number of elements

dition

C

K,

x,,

(*

>**) =

is said to be satisfied by a set

of

A

a maximal set

of

/\

R -by

R

by which

C

R

,

C(x

j

T(.^^

for

This con-

K).

1| 2,...,

is satisfied for every finite number of elements

If there is a set

be a condition

A (f =

(

9

1*

...,

*)

2,...jx).

is satisfied, then we can find

which

ff

is satisfied, that is. there is

A by which

ff

is satisfied* (C.f . Introductioh

in

Volume II or III in this Series: H. Nakano, Modern spectral theory,

or

f(

of

other set including

po

Topology and linear topo logical spaces.)

For two spaces #. and 5

,

a mapping at of R, into

a one-to-one correspondence between

elements

X

,

R

,

the images

R

and S

,

3

is said to

tie

if for every different

and 01 fy)

are different, and

3

for any

no can find *

*

to-one correspondence Ol between

*

a

R for which and

R.

<S

we also obtain obviusly a one-to-one correspondence between

4

This one-to-one correspondence Oi"

by

1

R

and

.

is called the inverse of Oi and denoted

Concerning the inverse

.

one-

# = *(*),

for

i(})

For a

Ol(x).

putting

,

0l

-l

we have obviously by defini-

,

tion

X

=

<3C*(oi(x.))

If a set

R

of

/I

x*R;

for

= <&(<*"())

^

for #

.$.

A

is not the empty set, then we can consider

x

itself as a space of all elements

A

A

For two sets

.

B

,

dif-

ferent from the empty set, if there exists a one-to-one correspondence

between A and that

A

with that of B

easily that A

A

~S

i

B

~

5

6

and

considering A

as two spaces, then we shall say

nave the same density, or that the density of

B

and

~

8*

for every set A

^

and we write A

i

^A

C implies A

~~

C.

A

coincides

With this definition, we

A

;

~~ 8 implies

B

~A

see

and

>

Thus we obtain density classes, i.e.

every two sets in the same density class have the same density, and every two sets in different density classes have different densities.

We define further

density classes will be called cardinal numbers.

Cardinal numbers may be

the cardinal number of the empty set.

by

This

called often densities.

For two cardinal numbers

w

and

w

mean that we can find two sets A and

,

B

we define *

<**

3

,

number

w

such that

With this definition, we have obviously that **

-H-,

p < ?i for every cardinal number p

and

R of

R.

^ ** or ** ?

**

/4

$p

:>

5

6 'f*

implies

and the cardinal due

Furthermore we have the following theorem

.

** to

<

to

Bernstein:

Theorem.

**,<*!>*'**!' implies If in.

Proof. ,4,

4*1

and 6^ c

^

^

tt

,

n,

such that

$ -HI, >4

*

= -H

.

D 6,

DA.

Since

>4

exists a one-to-one correspondence Oi between A 9 and A\ set

X C Aa

,

putting

^

then we can find by definition ,

X,

^ For

,

there

each

4

we have obviously

C A<*

(x)

01

A* = 0l(A^> we have

^ D

^>

>4 a :> J3

t>

v = 1, 2,...

B. =

ft

Vat

-

^

2,..,

a*^

B|^^ 4 &,.}>

t

f r " ^ 1

0t(B--i)

&*

<7t( B,,..,

for every

Thus, putting

)

Since

.

(&-,*--..{ i.3|

/*-)

+

for

x

51

(

for

x

2^

(B*-t->IJ

BJ},

(/<.--

putting 01

(x)

*

between /^ and

we obtain a one-to-one correspondence

hence

-fv

If a set

of

x4

K.

is infinite, that is, if

A

we shall say that the cardinal number of K.

B,,.

8,

and

,

= -H

number of elements, then the density of

A of

A,.,-

A

A contains an infinite

is said to be infinite,

For a set

is infinite.

if there exists a one-to-one correspondence between

,

the totality of natural numbers, then the density of

A

or

A

and be

is said to

COUR table, and the cardinal number of the countable density is denoted

Theorem.

is the least cardinal number among the infinite

^o

cardinal numbers.

^X

A

Let

Proof. (

ax

be a set of an infinite density, and

C Rj be a mapping by Choice Axiom.

For an element

X

<X 6

A

,

putting

we can find by induction a sequence

have by definition

A^

Let

x

\

X

(

{

x

A

A ),

$ )

5

OL,,

*

A (v =

4 for the cardinal number C^

be a system of cardinal numbers.

we obtain a sj.ace

Thus we

1> 2,...).

A of elements

of

A For

a

set

(A

,

for

*)

&

If there exists a set A of

respondence between

A and A

A

x

%

,

e

/4

X-

such that we can find a one* to-one cor-

^

then for the cardinal number

,

of

/)

we shall write

If

X

R

is a set of

/I

A

6

,

with the cardinal number

A and

x = <W for all

then we shall write

With this definition, we see easily that **,w> = /fm*, and i*f,(

4*

for cardinal numbers -W,

)

^

Theorem.

=

.

,

\

for every infinite cardinal numbey

Between two spaces

Proof.

i*

=

(*****')

(^,^*=1,

(i',/*)

\

and

2,...)

(X = 1, 2,...), putting

^-

-

1)4- k

4

(i/,

=

1, 2,...

Thus we have ^ 9 1> =

we obtain a one-to-one correspondence.

^

fl

by

definition. If a cardinal number

is infinite, then for a set

find by Maximal Theorem a system

#Q

A^

A * A tA

(

=

A*A?

A space

X

R

i

plies

it

space

R

R.

x.

^ Zjand x-^J,

< y.

^ O -HI-, and hence

^ x such that x < A implies x = ^

or ^

^^X

.

is said to be well ordered, if every

A

=s

T

is said to be linearly ordered, if for overy two .elements

it is defined

the minimum element, that is, if there exists

every X

x*f.

for

Thus there exists a cardinal number -m for which

we can

sucn that

A)

*-

A

j

,

^

Z.

im-

linearly ordered

?^

A

ex

A such that

*

^

of

R.

contains ex 5=

X for

.

Zermelo proved by Choice Axiom that every space may be well ordered.

We shall prove now however this theorem by Maximal Theorem. nite set may be obviously well ordered. well ordered sets of

R

.

Every fi-

We consider the totality of

For two well ordered sets A and

6

,

we

is a section of

shall say that A

valent to

-

B

J tem

X c 6

if

>

x,

/4

#

x < # in A is equi-

;

x

and

;

x/4,

for

By virtue of Maxi*ial Theorem, we can find a maximal sys-

A

W,

y

=

in 6 for every

3

R

of well ordered sets of

flt

sets of

^

*

8

subject to the condition: for every two

one of them is a section of the other.

X

such that every X

we obtain a well ordered set A

,

putting

Then,

WC

is

X77t a section of X

Furthermore we have A - R

.

then for an element

a well ordered set

tain

W.

R

Thus

R

Let

set

i

A

,

a

a}, and it is evident [A

{/A,

of

A . putting x < & for every x

R

a.

A $ A

ty with

,

such that every X

}

M

6 T/T

R

R

We shall prove that for

R

any

which has the same densi-

A

is dif-

We consider one-to-one correspondences

.

between a section X of

R

such that for every

^x t

tflxti

^x,( x ) =

section of the other and

^x

t(

ft x

A

By virtue of Maxi-

Trr

of one-to-one correspon-

and a subset of

mal Theorem, we can find a maximal system Ct K

a section of

7/t is

We can assume obviously that the density of

.

* R>

we ob-

,

contradicting the maximality

,

there exists a section of

ferent frorc that of

dences

/4

may be well ordered.

be a well ordered space.

of

Because, if A

.

*

^>

x)

one

X

f *t and

f r every

x

z

a

is

For

X, Xi.

such a maximal system W? , putting

=

X

5". ft**

W

X,

(x) =

ft

we obtain a one-to-one correspondence

A

subset of

and we have

,

Xtf

the density of A

If

0i

Xp

(xJ *

^{X,, *}(*) and

01,

A

=

.(X,,) >

a

=

of

R and a

6

OL

/4

R -

,

^Xo

for the minimum element X

X

*R

X ')> putting

(

for eveiy

^X/*)

Since

we have obviously

x ^

Xa

R-

X<,

of

we obtain a

,

*oi

one-to-one correspondence set of

of a section X

Xo

is different from that of

then for an element

/4,

0i

*X,

for

x (x)

by the maximality of #T.

* Trt

tfi

0l

X, and

Thus we have

For every section X

- X

,x #

,

=*

R

Oi * v *0

R

we have obviously

Therefore we can state

_ ,^ftt

,

^ = of

a sec tion*

J^

it is obvious that

mality of ^T.

ment of R

^^x

tfl

,

I

^f

{

X

,

x,

f

of

R

and a sub-

contradicting the maxi-

x (X d ).

denoting by

X K ^ Xy

X

x

the minimum ele-

if and only if

X C Y.

7

For every space

Theorem. in

/^

#

the totality of cardinal numbers

,

is a well ordered space.

Concerning topology, we shall make use of notations and terminologies in Volume II or III in this Series.

But, in this text,

we

need only be aquainted with elementary properties as stated in Intro-

duction in Volume II

Modern spectral theory.

:

Concerning complex numbers, we shall make use of usual notations: for a complex number

d,

the real part; by

ft
the conjugate complex number; and by

o?

imaginary part; by

we denote by

,

*yo<

|oc|

the

the ab-

solute value, i.e.

X let

I

=

K*

=

{(*)*+

+

CM

I

#<* -

=

3

9

i 7<*

*

CfoO* = v/^5

The space of all complex numbers is called the complex plane and

A topology is defined on the complex plane in usu-

denoted by Gf.

al sense: the system of point sets

{

$

: |

?

-

o(

<

|

>

for all

I-

&

o<

,

*

constitutes a neighbourhood system. Let

R

complex plane

Q- is

called a function on R

lue c?(x)

is real for every

A function

f

any

>

A mapping

be a topological space.

on

R

X

R

!?(*)-

^

A

*

*

nuous at every point of

is said to be real valued,

*

X*/l.

for every

^ C

/^

,

if

(**) = oo

,

is said to be continuous at

can find an open set

A

*

*

| considering

|

oo

|

= + oo

.

^

is conti-

We may often consider a function

which takes the infinite point oo as a function value.
#, if for

such that

?(*)!<

A

into the

If the function va-

said to be continuous in a set

i9

R

of

is said to be continuous at a point

we can find an open set

A ftuiciion

then

,

X

fl

>

In the case:

if for any

such that for every

x

<-f

/4

,

>

we

HILBERT SPACES

Linear Spaces

1

A space

4

is called a commutative group, if for every two elements

R.

we have a, +

e /I

+4

4

$

such that

&

-

l

0.

2)

(<X-f)-t- C

3)

for any

&

=

4

,

R

Such an element C

(^-t

4-

R

we can find C e

+

C

C'f

t

C' =

o(,

-

-

C

.

Because, if

',

then we can find by the postulate 3)

=

such that O, =

ft

is uniquely determined.

= ^ + C =

<X

C ),

0-

d

R such that

*+

C,

and we have by the postulates l) and 2)

= C'^

fl(

3 C

.

R

Therefore we denote such C 0.

Especially

=

^

-t-

by

a.

a- A for every

a

^.

and denoted by

-a.

<+

.

l) and 2)

obtain the same element

6 8, we

a.-

a ^ a

for every

-

61

^ ft

,

a

This

.

R

is called the zero element of

<x

t

-

of a commutative group /^ we define

(- Cu) = (-

a)

+

(-4)=a--

=

-(-a)

<X,

a

4- (7

for every

=

(X,

(X,

we nave

4

R

O.

to

because

(-(-*)).

(cx+ (-<)) =
a +

K

6

+ (/-/).

(^ -/)

With this definition, we have = 0.

Since

^

0.

For each element

Q

it,

we conclude therefore

= ^ -

uniquely determined element

mean

Thus we have

.

for every

(a,- 4)

-(-/*(-^))+

,

-^

^.= ^-t-(^--^), and hence we have by a <* + <-*-*)) + (*-/)

By the uniqueness of

that is

<X

.

Linear Spaces

l)

A commutative group and complex number

R

is called a linear space, if for every

we have

o<

9

R

o< a.

d.

K.

such that

4; 5)

6)

la

7)

Since OL

(X

fl

*

(X

Furthermore

Let

we have by 5) and ?)

* (-l)<X =

Oa

((X- ^

s

)

=

and hence

,

* <x

-

^ for every

of

=

ot(cx + (-4))

R

,

0-

,

,

4

t

A implies

o/a^< (-1)^

*t fr

+

for every

R

4 s

o(

t

because

a-o/^.

A set of R is called A manifold A is said

if it is not the empty set. (X

& =

(-l)a * -

be a linear space in the sequel.

to be linear* if o<

f) *

(a-

<*

R

by the postulate 3)1 we have o

Therefore

a manifold of

numbers

&

=

0-

.

o<

= a.

A for every

f>
complex

.

,

jS

For a manifold A combination from

At

and complex numbers

R

of

an element

,

a

R

is said to be a linear

A

if we can find a finite number of elements x*, <^^ (i/

=

2,...,>t) such that

1,

*.

^1 t-af

a =

<*,*,,.

We see easily that the totality of linear combinations from tutes a linear manifold of

R

This linear manifold of

.

A

the linear manifold generated by

R.

is called

It is evident by definition

that the linear manifold generated by

including A

A consti-

A

is the least linear manifold

.

A system ax6K.(>-6/\)

is said to be linearly independent, if

every one is not a linear combination from the others, that is, if for a finite number of elements

X^tA

and complex numbers

c^ u

( t-

=

1,

2,

-,*) ,21.0^ i-s

ax =

implies

.

o*,,

=

fftr

every

v/

=

1, 2,

,*.

I

Functions on a linear space

functional

^

on

R

R

are called functionaj.g on

is said to be linear, if

R

.

A

HILBERT SPACES

10

for every elements X

,

R

jf

V

For two functionals define a functional &
(Chapter I

and complex numbers

V

,

on

R

c*

,

ft

and complex numbers

erf

,

we

,

as (Jif

(<X
oi

<jp(x)

+

(S

V

for

(x)

xfR.

With this definition, we see easily that the totality of linear func-

constitutes a linear space.

R

tionals on

A functional

R

on

ff

for every elements x

is said to be anti-linear, if

R and complex numbers

,

is anti-linear, if and only if, putting

^

we obtain a linear functional

on

R

(x) =

c*
,

,

that is,

With this definition,

.

x

(x) for every

also see easily that the totality of anti-linear functionals on

R, we

R.

con-*

stitutes a linear space.

Normed Spaces

2

Let

R R

a norm on

,

)

0$MXii<-foo

2

)

H

4)

II.

*X

||

X+

llxit

-?c

II

(

II

*

R)

is called

if

1

3)

A functional

be a linear space .

-

II

|

o<

|

II

~

}t

X

X

for every

x

II

R

,

for every complex number

il

4-

II

II

,

,

x =

implies

cX

.

A linear space associated with a norm is called a normed apace. Let

R

lirn^

if

lira V >

A sequence (X^e

be a normed space in the sequel. is said to be convergent to a limit

1, 2,...)

fl

a, -

aft

-

R

<x

If a sequence

a^

6 R,

^=

=

,linJia.,-4!!

1, 2,...) is con-

Because, if

0,

by 2), 5), and hence

a

2=

^

by 4).

Concerning convergence, we can prove easily Theorem 2.1.

J

(^ =

a, = a,

vergent, then its limit is determined uniquely.

^limja,,-*!! then we conclude Ha.- ^H *

R.

and we shall write

Urn

a.,

= a,

lim^-l,

s I

,

then we have

Normed Spaces

2)

for every complex numbers we have then f

o*

oX +

$4

If

.

,

lim *

=

0*0,,, 4 j&A,)

lim

11

(3

lirn^

a^ =

&

lim

,

=

ot^

o^ ,

o( ft.

H -ow o<,,a.,,=

From the postulates 2) and }), we conclude at once I

*

H

-

I!

II

#

II

x -

II

I

II

.

There to re we have

Theorem 2.2.

A

A manifold (i/

=

R

of

implies

(X

A

6

6

,

A\ also

xp

=

-fo( k a.(*/

oi k

l, 2,...) is convergent

x

c* ^

/4,

,,

(

i^

If

(*.>,

^lirn^

+

subsequence

,

6 also is closed, the intersec-

),

A

of R, if

a

>4.

2,...), then both sequences

1,

K.^

= ^,

x

A

fr

(f = 1, 2,.,.)

lim Jo^

|

s

-*

<x>

c>^

o^"i

= 1

( i/

A

and

*

/4

Therefore

is closed.

,

c*^

1, 2,...), putting dL ^

=

lim

i/^c

o^ Ml

.

/^^

*

, '

r

ofg

lim

^-*o

t

, '

we have by Theorem 2.1

>)

is a closed linear manifold by assumption,

X

5 .IjJ".

and consequently <^ w (i/

a

2,...) contains a convergent subse-

Furthermore, for every two convergent subsequences

quence.

then

for

(^u=l, 2,...), then we have by Theorem 2.1

<^^

every subsequence from

Since

A

a** 6

for a sequence of elements

Because, if

contradicting the assumption:

V,,

=

^a)

(^ = 1, 2 ,.<) is bounded.

(^

(X

(i/= 1, 2,...) are convergent.

Proof. el,,

a^ =

4-

A

A^ (x

For a closed linear manifold

and complex numbers and

<x||,

is closed, if it is not empty.

Theorem 2.J. and

fl

lim

the union A

and that for any system of closed manifolds

XM

=

With this definition, we see easily

.

that for any two closed manifolds A

tion ~T[

lifjlaj/

is said to be closed , if

a

implies

2,...)

1,

a. =

^lirn^

<*

V=

(S

,

^

*r,,

because

a

- (P - *) ?

convergent by Theorem 2.1.

obtain hence

,

Thus we conclude that

/4

= 1, 2,...) is convergent, and hence

ire

X^

/I

(^ a 1,

,

HILBERT SPACfiS

12

(Chapter

I

Prom this Theorem 2.3 we conclude immediately If a linear manifold

Theorem 2.4<X-

? A

the linear manifold generated by

i

A is closed, then for any { <x

,

A

\

is closed,

We see easily by definition that the intersection of a system

closed linear manifolds also is a closed linear manifold.

cluding

A

A

For two manifolds B C A of if for any

,

such that

lira * **

-

R.

,

d.

*

6

Let

lim

^

/4

Since C.

C

1, 2,...)

is

C

and

,

the

= X for some sequence

C

x

R.

^B(i^

subject to

=

l,

2,..)

is a linear manifold includ-

C

is closed.

must be included in C by definition, we conclude hence that

-C. A sequence

if for any

<X

>

R>

t

lim K,

(^=

1

-**

||

a,,

-

a,*

2,...) is said to be a Cauchy sequence

,

we can find n <X *

that is, if

such that

j^ ||

~ -a^H-

for

$

*

,

/*

3 4f

It is evident by definition,

#

^V*.

that every convergent sequence of elements is a Cauchy sequence.

manifold (is

A

It is evident by definition

.

Furthermore we see easily by definition that

.

in

is dense

=

A

the linear manifold generated by

Then we see easily by Theorem 2*1 that

5

(^

/.

Let C, be the totality of elements

.

the condition:

ing

B

The linear manifold generated by a manifold

closed linear manifold generated by

B C C

t

With this definition, we have

dense in the closed linear manifold generated by Proof*

8

we shall say that

A we can find a sequence 4*

<x

Theorem 2.5*

that

in-

This least closed linear manifold is called the closed

linear manifold generated by

A

Therefore

there exists tne least closed linear manifold

A

for any manifold

of

A is said to be complete

1, 2,.*.) ia

.

if every Cauchy sequence

convergent to a limit contained in A

O-^

A A

*

With tnis definition, we have obviously

Theorem 2.6.

Every complete manifold is closed .

Furthermore we see easily by definition

Theorem 2.?,

For a cororlete manifold

A

and a closed manifold

Normed Spaces

2)

B

ia a complete manifold, if it is not empty.

the intersection

AB

Theorem 2.8.

If a linear manifold A is complete, then for any

,

Q ? A the linear manifold generated by

For

Proof. (

13

=

x*

=

>| (is

{ <x

Is.

/( V

,

complete.

andctXif

1, 2,...)

X,,

+

*

a.

a Cauchy sequence, then ve have by definition

1, 2,...) is

Since A is closed by Theorem 2.6, ve conclude from

A by Theorem

0c

2.3 that

lim

A

Since

(

X* -

s

xp

lim

,

^

(<**,

-

(7 .

oj^)

is complete by assumption, ve see that both

x *>4 for which

are convergent, and there exists

K^ lim

A and

<*V

* = x

Thus ve have by Theorem 2.1

oC00 = x

lim (x* +

-f

(ijjn^ot^)

(V

.

From this Theorem 2.8 ve conclude immediately

Theorem 2.9^

The linear manifold generated by a finite number

of elements is complete.

For a sequence

21

&

,(" = 1, 2,...), ve shall say that a series

is convergent vith sum 0u, and vrite

0~\,

=

OL

if

lim

^".. CL.H

A.

2^-

Hr

fl-i" >

I

With this definition, ve see easily by Theo-

rem 2.2 99^

0*>_

if

x>

Ot^ is convergent.

If a normed space

Theorem 2.10*

21 then

21

&i/

If

j?L lim

tl

Since

R

2I 21

^

CX^

(i

< +00,

is convergent.

Proof.

Since

H

is complete and

| II

22-

(I

a^ ||< + oo, then ve have

^

22. H O. v

II

>

.

U

we obtain further

is complete by assumption,

^L I'Xl

Ct v

lim

H

X ^

II

is hence convergent.

^

HILBERT SPACES

14

Adjoint Spaces

3

Let

8

(Chapter I

An anti-linear functional

be a nonaed space.

ff

on

R

is said to be bounded, if

sup

x%1 Denoting by

R

(x)

|

-c

I

*

oo

.

R

the totality of bounded anti-linear functionals on

we see easily that

R

constitutes a linear space.

,

Furthermore, put-

ting

Hall

(1)

we can prove easily that

space

R

R

fc

ll

^j

(aR),

|*(x)I

sup H

I

This norned

also is a normed space.

R.

is called the adjoint space of

R

Concerning the norm of the adjoint space

R

of

,

we have obvious-

ly by the definition (l)

Hall =

(2)

J^i

sup

U*R),

J|X H

and hence we have (5)

|

Every bounded anti-linear functional

= X lim V-00 X^*

lim

implies

is continuoua. i.e.

I,

because we have by the formula (j)

|y(xj- ?(x)|g imiiix,,- xn Conversely) if an anti-linear functional
is bounded.

Because, if

definition a sequence

B*.,lll,

x,

i, 2,...).

then

R. (^

l*f(Xv)l

d/

is continuous,

=

g

1| 2,...) such that

2

k

(v = l, 2,...),

and then we liave ^lim^

-jp*,,

contradicting that

Theorem J.I. Proof.

= 0,

iy(-p

^ISi

(i/= 1, 2,...),

is continuous.

The adjoint space

If

X

liy^na^-S^a

R

=

o

R

is complete..

for a sequence a.,*

R

1, 2,...), then we have by the formula (3)

^UmJO^x)--

(*)!

=

for every

Thus, putting
(x)

=

lim

Cl^X)

(X

*

R),

X

6

R.

(^ =

Adjoint Spaces

3)

we see easily that

is an anti-linear functional on

*f

more we obtain by (5) for every

r (x)- ?(x)(

|

15

*((

|l

=

/u

lim

||

Further-

.

1, 2,...

for every

x

*

is bounded, that is,

6

5^11

a^-

From this relation we conclude that

R

.

R

,

and we have ll

ar -

i^

H

5^

-

(/*= i, 2,...).

This relation yields lim

*R

A) 2n H, i

(X

sup A /\ and

R.

civU)

|

<

I

-f

oo

x

for every

R

,

is complete, then we have

sup

/|

** A If

Proof* "

Xsup 4 A

"

X^

II

For a system of bounded anti-linear functionals

Theorem J.2. 5L X

ZL*-
/|

^

is complete by definition.

Therefore

A and X^

||

(*=

R,

ax aA

II

< +00. =

||

t

<*

,

then we can find by definition

1, 2,...) such that Z" 2

IS^UJI^

11X^11=1,

(/=

1, 2,...).

Then, putting

&

=

4^

^

^w =

-jr ^i/

(v/

=

i,

2,... )^

we have I

(*>>!

!

lirn^

^(x)

ll^tf as

-

^(^y-) Therefore we can find a subsequence lija^

Uy

(^,-t-

i-L,
because

R.

for every

X

for every

v/

^

(/u-= 1,

^%. )l< l

1

l-J^

-c

f

lj ****

For such (a.,

1,

2,...

.

2,...) such that

for

^ = ^T^ \\a^ |j<* V" 4^ 2 /*t r^ r^T

Jx(>l-

R, =

-y,

i^

is complete by assumption. I

(w = 1, 2,...)>

-JTT

^^hence we can

ex.

^

8^

...+^.,)l

we have

I311

*

(Chapter I

HILBERT SPACES

for every

At l

=1,

A sequence

R t

f*-

0.

(v = 1, 2,...) is said to be weakly convergent

R

a,,

a

to a limit

1^(^)1 r

lim

2,..., contradicting

if

,

lim *

s Z (*)

(*)

a,,

*^

x &

for every

,

and then we shall write

a, s a.

w-i^

la k (x)-a(x)ls

Since

lla,,-0t

lim 5^ = -4**

a

II

Hxll

by (5), we see that

w-lim V >"

implies

|r

=

(Xj,

a

Furthermore we conclude likewise "

-

"

'

"" 1

=

lim

Theorem 3.3. 1

^

-* ^lim^

Theorem 3.4.

%

For every

Proof >

fl

w>.*

"-*

II

lim

implies

lim I5^(x)|

Thus we obtain by definition

.

a (a).

=

a-

implies

O,

a^ll

^

lloLll.

ve conclude from assumption by (3)

6 R.

|o,(x)| =

= lim v-*o a^

,

a^aj

5^ =

*-lira t-**o

r

'

^

^

lim

ii

r^5To 5,

II

^

lim

||

cL

II

.

r^rs.

Theorem 3*5-

implies

lim

implies

R

If

sup

ft

a*

<

(I

II

is complete

< + o*

lim^

enifold

A

.

then we see by Theorem 3*2 that

Since

a,,- alllia^n +

we conclude hence that

A

is complete, then

= 5(0c).

<x k (<*,)

Proof.

R

I

of

I

a,,

(a)-

= 5 v-lim^ a+ lij^ a^CaJ * a (a).

X

,

(X^

a

(a)

=

a

A

coincides with the whole space

With thie definition, it is evident that if a manifold

&,

then

A

,

implies

is said to be fundamental, if the closed li-

near manifold generated by

in

I

is fundamental.

A

R

.

is dense

4)

3,

Adjoint Spaces

Theorem 3.6.

5,,R.('lf2 every X

For a fundamental manifold A of ..), jif

f

and

/4

17

av

|{

^sup

||

<+

a^ (^s

then

<x>,

for

is convergent

(fsl,2,...)

(*)

5^,

and a sequence

R

is weak-

1, 2,...)

ly convergent*

Proof.

Denoting

6 the linear manifold generated by

f>y

see easily by assumption that

E

every x R.

Since

by Theorem 2.3*

(^*

k

=

^ putting

lirn^

1, 2,...

x^

R,

fc

= x

l(

,

and consequently

(

X^

s

/*

1,

we obtain thus,

Making

^(x)

is dense in

Since we have by asaumptlon

.

1, 2,...

/M,

r

s

II

X - xr

tend to oo

ft

,

A

we conclude hence

l t 2,...) is convergent for every

x

Therefore, putting

we see easily that

0-^

(^ =

W| S%|I^

*

g.

A functional

be a linear space.

(*,)

2)

(X+ y,

=

(<*#)

4)

(

(^^

f

^

- n

(

4

mnl A <(

(

*R)

x,

x,^R,

y,Z), for every complex number

for every x

,

(x, f)

if

for every

<*(*,)

X x) J V

,

Z) * (X, Z) t

3)

*

R.

(**) =

and

,

r-

Inner Product Spaces

is called an inner product on 1)

R),

R

is weakly convergent.

1, 2,...)

4 Let

(X

lin^ a^(x)

is an anti-linear functional on

H
*

(x)

we

Then we have

.

a

for every

&

we can find a sequence

for evtry H a,,

,

convergent for

1, 2,...) is

(^

,

=

If

5L k (x)

is fundamental by assumption h

Thus for any x

such that

1, 2,...)

for every

/4

>4

.

e

R /)

,

c*

,

&

B

HILBERT SPACES

18

(Chapter I

Concerning inner product*, we conclude from )) and 2)

+ *) = (x,

(*>

2')

+ (X,r.),

and from l) and 3)

fr

(*,*)-*(*>#)

V)

very complex number

o<

.

A linear space associated with an inner product is called an inner product space*

Let

R

By virtue of the

be an inner product space in the sequel.

postulate 4), putting

*

11X11

(1)

<

*e have

X

II

<

II

(xR),

\/(x,x)

+ 00, and we obtain by 5) that

x

-

^^

implies

II

X x 0. For o *

R

t

jj

putting

,

\

I

wa

i

for

jj

Thus we conclude by the postulate 4)

for every

!(x, f)ll IIXMIIJH

(2)

By virtue of this formula, we obtain

and hence

Y * Therefore

M

x

fl

(

x

H

R

S

I

X

H

II

J

is a norm on

)

R

as the norm of an inner product space

II

.

R

.

This norm is defined

Concerning the norm of

.

we have obviously by definition *

X *

=

fl

H

l

X

dt

2R( X

,

)

+

||

II*.

Consequently we obtain (3)

ft

x+

p

1

+

2

2(HX|| +||^M

Hx-jB*

1

).

Purtnemore we have

XU.j)-

1

-J-nx'+^i

-

1

-J-iix-^ii

Prom this relation we conclude

^(*,^)

=

Therefore we obtain

KU,{)

=

-i-

BXH ^u'-iil^-

l J

#

||

.

R

,

Inner Product Spaces

4)

19

1 (*,$) * ~{(*x+}f-*x-lfl + i(l* + i)f-lz-l}l )

(4)

Every inner product apace

is a norroed space ty the norm of R

R.

.

Conversely we have For a noraed apace R

Theorea 4.1.

,

if the none of

R

satis-

fies the relation (}), then there exists uniquely an inner product on

R

*

(*,)

II

X

R

e

*,

(

H

V

at

such that

)

(

x, x)

For the norm

Proof.

(*#)

for every

x

II

t

=

--j

j-{(**+n

R

of

||

1 )

,

R

6

.

putting

cdia

+

z

+

1 ) J

i^-iu-ijii

we have obviously (

x,

s

x.)

x

R

1

for every

U

X

R

*

.

Furthermore we see easily that

U*y)s

i(x,),

For an arbitrary complex number x

^

(x, ,

{*.

)

= -

(x, j).

we have by the relation (3)

(x+ x )(x,^). x =

Therefore, putting

(otx,^) = (x,o(^)

and hence

quently, putting x =

<X

k

tf)

X

for every real number

for an arbitrary real number

(o*xjf) = *^(x

and hence

^

for an arbitrary real number

i<*

<X

for every real number

,

we obtain

Conse-

.

we obtain

,

<*.

Thus

.

we conclude now that (

X

x

)

,

s X(

x~

,

^

for every complex number

)

x

.

By virtue Of the relation ()) we conclude further

(* Therefore

f

t)+

(jf

,t) a

j(x^^

(x,) (x ^
(

x x ,

}

s

R

x

2

,

2^) =

(X-*-^

is an inner product on

for every

x

R.

,

2.).

R

such that

.

The uniqueness of such an inner product is evident by the formula (4). Every inner product space by the norm of

R

.

R

nay be considered as a norroed space

Therefore we can consider convergence of ele-

HILBERT SPACES

20

ents sequence* in an inner product space

(Chapter I

R

We see easily by

*

t|if

formula (2)

Theorem 4.2*

lim, c*

*

=*

,

4* = < implies

11g

An inner product space is called a Hilbert space, if it is com-

nomad apace*

plete as a

Adjoint Spaces of Inner Product Spaces

fo

Let

K

considering

be an inner product apace and

as a normed space*

R

?*(*) =

(1)

R

the adjoint apace of

For every

|(

%,

II

||

a

is contained in

ya

on

/?

Recall-

*

4(2), we see at once

(2)

Therefore

R, putting

a.

(*),

(*,*)

we obtain obviously an anti-linear functional

ing the formula

R

for every

II

R

a

R

Furthermore we have obviously

.

ty definition

for every

0.

R

-

,

and complex numbers

Theorem 5.1. then we can find

*

X

(tx

=

*

a?A 1,

.

/(

R

of

if

,

a

a.

f

Jf

s

for every

(*)

inf AV ^

II

X - All,

we have

x

For an arbitrary sequence

K>

,

because X

^

/)

^

>

(^ = 1> 2 y *.*) such that

(^ = 1, 2^*.*) such that

putting 58

>

^.

2,..*) such that

ctiT

X

We can find obviously a sequence

by assumption.

Then we can find further a sequence

we have

?

such that

R

Jt 6

Putting

is closed and

p

,

For a closed linear manifold

5. (a)

Proof,

<*

-^(x^-a^)

(p

-

1 9 2,...),

Adjoint Spaces of Inner Product Spaces

5)

1

IUvi,+

21

x*,

(\

On the other hand, we have naturally

^^X

o< v

P6

1

*-

Kftyl^HJU*,, -a,J>) + < I? ** - o-u 1 a 3^-2 M(x>, - a ,}) I* t *

-a)!!

(x,,

a,

a u1

-

(^ = 1

/4

il

2,,..)>

hence

.< #>

** -

-t

g r-

flutt

(^*

i, 2,...).

Consequently we obtain

2|(*i,-A =

^

for every

1

jMl^CUC +

f

1, 2,...

)!!^!!**

ar

t

f

This relation yields

.

Therefore we have especially

(?C^-a,

(

^lljn^

Since

lim

^lim^ x^ a, x) s 1

NX^-ail

^ a,

)

X

for every

o

6

A

.

= f 2 , we conclude hence

<*-., a).

.^

We obtain by the formula

-IT*.

4(3) for every

^

,

ft

=

1, 2,. *

1

because -i

(

X

-

fc

A

6

xp

Thus we conclude

.

and hence by (2) and (3)

lim W,^-

n /x x i*
y

Since the adjoint space

?

ists hence

ting

OL

=?

x

(

-

R such

ya

,

II 11

*

'Hp*

R

of that

lim

is complete by Theorem J.I, there ex-

?x ^

X

=.

For such %

.

we have by Theorem J.3 for every

x)- (a,x)= a.ci) = - X*

^

^lim^

*

%

(x^ - A

e

R

,

put-

><

9

x) *

(?,

o.

Thus our assertion is established.

x

For any

i?

/ we see easily that

a

an element

f

R,

(t/

s

t R.

t

s

A

putting

{X

X

:

(x)

=

\,

is a closed linear manifold of

such that

X

is 2,*.*) such that

(a)

+

,

R.

then we can find

If there is

a* R

And

X

fe

HILBERT SPACES

22

5 (*,) $

(Chapter I

R (X)

,

X

for every

x

Then for every

as proved Just above.

A

,

we have

R

*(*)

0, that la,

x(a)

JL^l

5(x)

5 (a)

we have

,

Therefore we have

Theorem 5.2.

For every

X

:

{
R} is denae In the adjoint apace R*

x^fR

Cauchy sequence

-nx r

IHXUI

&

iil

Purthermore, for every two

xw -

n

(xv>

2,...), the number sequence

for every

jx

-

*)

(*i,. ,

=

M.

(

V

JI

!f/

1, 2,...

the number se-

and 4 W

<

(^ =

^)

*

X"

{l

l(l1

^-

X

R

e

J

,

^

* i, 2,...)(k = 1,

1, 2,.*.) is convergent,

J/--

11

*

tt

*~ V"

I(

^

lim

'

such that

y Tv

!,->

J.

Jf*-

By virtue of the formula (2), we see that such sequences x k and (v =

1, 2,..) are Cauchy sequences*

(* we see easily that

adjoint space (4)

R (

,

because, if

^lirn^

x

|)

,

)

jj

Thus, putting

^^(xv, ( 5c

f

>

^

fat

S)

fJ, is an inner product on the

and furthermore we have

?*

(X

(5)

(

f

H

we can find by Theorea 5*2 two

>

(^ = 1, 2,...)

R

9

^ R

and

X,,

4(2)

* x, '*^ !,-* yllm

(p

n

.

For every two elements sequences x^

x^

Cauchy sequences

because we have by the formula I

(K = 1, 2,...)

(^ = 1, 2,...) is convergent, because

quence HX^N

,

,

^

TL)

f x^

m

U

)

II

j)

4

S

= x

,

/|

4

then

for every

for every X rfe

x

R

,

^

R,

,

have

xj

s = = ( x , x). 11*8* = ^lia^ lin^ ( x. , II^B*^ JLlin, Mac,, I Thus we have proved that there exists an inner product on the adjoint

space

R

subject to the conditi*f>?

^)

and (s).

We can conclude

R

Adjoint Spaces of Inner Product Spaces

6)

5>

further ty Theorem 4.2 that such an inner product on

apace

R

aa an inner product apace

With this definition

Theorem

53

ft

associated with such an inner pro-

have hence by Theorem 3.1

.we

If

anti-linear functional

R

is a Hilbert apace, then for any bounded

on R

we can find

,

R

If

Proof.

R

x

is dense in

I

=

R

a

Therefore, for any

21

*

:

*

R

T

R and &

R

between

the range of

T

.

R

into &

such that

$

to

R

x

for every

.

Completion

A one-to-one correspondence

of

$

and

4

A ,

Tx

:

A

(

called

*), is

T

then

,

A transformation

.

x

is called the range of

coincides with the whole space $

R

Thus we have

5*?2.

R

fit

(-,%)

(*)

On the other

R K

X

:

we can find

]?

and a manifold

transformation from

R

e

TL

R.

of

j?

be two linear apace.

a transformation of

such that

}

R by Theorem

{
6

Let

R

*

is complete, then we see by the formula (2) that {
:

a

for every

is a closed manifold of the adjoint space

*

is

namely a complete inner product space.

Theorem 5*4

i

is determined

The ad.loint space of an inner product space

a Hilbert space

hand

R

Therefore we define the adjoint apace of an inner product

uniquely.

duct.

23

T

If

called a

i

R

of

T.

into

S is

said to be linear, if

for every spaces

then we

R

x

R

}f

and S

,

say that

and complex numbers

cat

For two linear

.

,

T

if there is a linear transformation

R

is isomornhic to

& by T.

from

R

to

t

With this defini-

tion, it is evident that the range of a linear transformation is a li-

near manifold, and for a linear transformation isomorphic to the range of

For two normea spaces is said to be isometric, if i

Tx

S

T* R and

T =

>

,

T

of

R

a transformation

into

T

of

is linear and ii

x

I!

for every

x

t

R

.

&

R

,

R into

is

S

HILBERT SPACES

24

R

For two normed spaces

ation

T

from

R

If both

R

then

,

is said to be isometric to

by

are inner product spaces , then we see by

T

R

of

T

.

the

into

*S

have

(Tx,T^) R

a completion of

T

R

R

R. into

ft

if

,

T

be

is iso-

With this definition,

.

the adjoint space

,

=

R

of

K

is a com-

(XR),

7*

we see at once by (2) and (3) in

R

of

yR. is said to

Because, putting

.

Tx ation of

T

is dense in

for every inner product space

x,

a Hilbert space

,

by a transformation

metric and the range of

pletion of

for every

(*,^)

For an inner product space R

in

I.

if there is an isometric transform-

,

4(4) that for every isometric transformation

formula \ie

S

to

and

&

and

(Chapter

3 that

into the adjoint space

R by Theorem

R

T

is

tun

isometric transform-

and the range of

,

T

is

dense

Thus we have

5.2*

The adjoint space

Theorem 6.1.

/?

of

aii.

inner product space

R is a completion of R by the transformation

TX

(XR),

m Cf^

?*()(*,#)

For two completions S, tively by transformations

,

T

t

S t of an inner product space and 1\

of T,

to that of

(

X^

=

(

A

}

Ti*

Let

1| 2,...)

(

1, 2,...)

respec-

(x
is an isometric transformation from the range

Since A\ is

respectively. find

R

putting

,

TlUx). Tx we see easily that "T g

(***)

A^ be ranges of T, and T* dense in for any X * 5 1 we can ,

and

t ,

such that

lij^*!,

* <

Then

Tp Xk

is obviously a Cauchy sequence, and hence we can find

T

s S z such that uniquely J Thus^ putting ^ s T* ^ lim^ $ *, we see easily that is an isometric transformation from S to S z

T

and we have

Tx

i

s T, x

for every

x

A\

Therefore we can

state

Theorem 6.2.

The completion of an inner product space is de-

termined uniquely within an isometric transformation*

25

Or tho normal Systems

7

R

Let we define

^ to mean

ex 0-

thogonal to ^

a

if

,

a

(

^

JL

4

,

=

)

,

and we shall say that

<XX

(2)

(XJ.^,aO.C

implies

(j)

OLA.

4 implies

oCaX^

implies

^

JL (L

aX4+

C

a-L^ implies |ia4.^|| = H&

(5)

&,,-*"

Zero element

,

for every complex number

(4)

+

l<*

a^ =

lio

2,...),

!i

,

is or-

,

l

=

ft R

,

<X-

With this definition, we have obviously

.

(1)

(

a

For two elements

be an inner product space.

ft
a.

oC

,

U*,

a

implies

J-

-

.

Con-

is obviously orthogonal to every element*

versely we have If a manifold A

Theorem 7-1-

is fundamental in

(OL,X) = (^,X)

a

then we have Proof*

=

Denoting by

(

a x ,

for every x t A

4

see by Theorem 2. 5 that

tion that

and

R

= (I

)

Theorem 4-2 that

X

the linear manifold generated by

5

5 is x)

dense in

R, and

for every x 6

,

= (^

X

=(a a-<)-U,*-
0,

)

,

)

we

Thus we conclude by

6

R,

,

,

it is obvious by assump-

x

(a

X

for every

Therefore we

have 1

fld-^tt and hence

ct

=

>

.

-^

For a manifold A of R

we define

,

aX X and we shall say that an element

Theorem 7.2. for any

a?A

A

to mean that t

A

,

is orthogonal to

If a linear manifold A of

we can find x

a-f xa.

/4

Denoting by

S

<*

Proof.

a

a

for every x 6 A

we see by Theorem 2.8 that

B

/4

R

H

*

a.4-

x

<x

x

is complete

and a complex number

>

if

o<

B itself as a Hilbert space and A

.

.

then

such that

l.

the linear manifold generated by is complete.

A

{a

,

A

/

Therefore we can consider

as a closed linear manifold of

5

Then we conclude by Theorems 5,1 and 5.4 that there exists an element

*

,

HILBERT SPACES

26 t

^

*

& such that 4

*

and

A

putting C = -~-r

,

fl< H

Furthermore, since we can find

X

we have C * 8

,

I

&(A

4

OL A

II

mediately

/* \xy

s

o<

^

=

^CA

"

1, 2,...,

x

since

,

C

K

,

x*

a

1.

/

}

,

c* a,

II

=1

6

(A

/I )

A

A

for every

is said

is linearly independ-

implies

Recalling Theorem 2.9, we obtain im-

Theorem 7.2

For a linearly independent sequence

Theorem 7.3.

we can find an orthonormal system

1, 2,...),

and

-L ,4

ax<1

R.(xt/t)

Kvery orthonormal system C x

for every

C

ft

auch that C =

An orthogonal system

.

Because,

,

such I

said to be an orthogonal system, if

i

/I )

to be an orthonormal system, if

ent.

?r

&) *

,

A and a complex number

for A ^ f

CL A J. CL f

^

/

is the linear manifold generated by { o,

ft

a*

A system

(

.

(Chapter I

C^

R. (*"

S5

O- I,6R.(*'

=

1, 2,...)

such that

C* =

JEL ^i/,^^*

**>>*

(^ = 1, 2,...).

*

Recalling Theorem 2.5, we conclude by the formulas (2), (3), (5) Theorem then

If an element

?4.

is orthogonal to a manifold

A

also is orthogonal to the closed linear manifold generated by

ct

For an orthonormal system

Theorem 7-5 U

1,

i!

2f

X

a

wt

Thus wd obtain

1

C<^CA Jl

II

x

ent elements X u *

=

1 tf

^x

A

A

for every

*

C*

)l

R.

(

)

^

A (f =

o( A

= (x, C x 1

II

^

XII

)

(x

we have

\^

A

k

A), we have

-

1

I^A^I

1, 2,..

M

for every finite number of differ)

f

and hence

It

x

a ||

j

T^

1 |

o( A

j

From this Theorem 7.5 we conclude immediately

Theorem 7.6,

/4

R.

finite number of different elements

...,K), putting

X - Zl

K*

> 2H

II

ver

For

Proof.

U*

d

For every orthonormal system C A

R.(\

A)

.

.

Orthonormal Systems

7)

and an arbitrary element

fL

*.

we have

.

x

(

2? ,

CA

a

)

UP to at moat

A

countable elements of

Per an orthogonal sequence

&*

A. (^

convergent , then for any permutation

fi

= 1

2,...), if JEL 0^^ 2,..., wT^.

fa,...ofl,

,

is 0.

p

f

i

also is convergent and

00

a

Because, putting

= 22.

&**

** nave ty

=

Ha-*

^e

formula (4) and Theorem

t

*-

2.2 1 <*

II

On the other hand, for any fi

contains

Ps

>

*

and hence

a

ZI

K

1, 2 ,

=

1 .

II

we can find

1, 2,...

X

. . . ,

such that f<

6"

Then we have by (4)

.

by definition. j-T. CL-,, ' Ir*|

a

dx

^ *A

R(x/1),

aA

Therefore, for an orthogonal system

= ^*.

to mean that there exists a sequence

...) such that

ax

for x

and then we shall say that

A,, (^

4e

T~ ax

*

X

we define

6/l(^ =

i/

*

a

l t 2,...) and

l, 2,

23 ^x v

,

i0 convergent.

JuM

With this definition we have obviously by Theorem 4.2 Theorem 7.7.

a

>T" 1 flail

o( x

=

For an orthonormal eystea

^(^^^)i

Cx

i

then we have

Cx

T!

I

(a,C A )=< x

^xl*i

X?A

for every

A

x

.

Furthermore we conclude by Theorems 2.1 and 4*2

Theorem 7.6.

a

=

o^ x

Cx

t

^

5x CA

,

l^x

Sk"*^* *

(

1

(X

=

ZH

<^x CA

i

linear manifold generated

S then t>y

I- *' <X

C* <

)

CA

1

(

A

/I

J,

ij

then we have

^1

>

If

R

For an orthonormal system c x

x

|o( A

pA

|

<

-i-

oo,

for complex numbers

Si'M

1

^

,

)***--

is contained obviously in the closed

&

(

A

*/l ).

Conversely we have

,

HILBBRT SPACES

28

a

For each element

Theorem 7.9*

ly complex numbers

of A

(

A

6

If we put

Proof*

/t

= (a

te number of complex numbers 2,

. . .

.

then for every fini-

+ Zl ***.- Zl *-! Wl *!..

A (^ =

\^

1,

1

1

l^

- Z-'I<*;U*. l^1

HCLii

I

is contained in the closed linear manifold generated by C x f R.

ex

we conclude hence by Theorem 2*3

)

Thus we can find a sequence A^

for x 4s

A)

A C\

-

ZL Mx.-l^l ^a

I

A

6

oi

{

"o-ii

I*

= 110-11*- r.l<x*,,|% V

(x

we can find unique-

)

a = TL (x

)

A

and different elements

1

*

ZI

i.1

Since

CA

*

we have

tt)

-

,

*

(

such that

)

of x

of the closed linear manifold

R

generated by an orthonormal system C A

(Chapter I

(f = l, 2,...)

A

such that

of A

=

(v = 1, 2,...), and then we have

A,,

ifa-

o^^cx, "

Therefore we conclude

OL

Theorem 7.10.

(

T~

x

/I

of x

ac

a^

hence

3^

I

o* A

C\ t _^r

1

<

I

*

A"

*"

& ^

^

(

1 v

I

*

A)

< +00

we can

JEI

^*<

o< Xw

-

a,

*

5H

*<x

*

be complete, it is

implies convergence

^^^

x^

^

A

v^ Cx

is convergent, and hence

CA^

a sequence

1, 2,...), and

Thus, if the closed linear manifold generated by

plete, then

1

is evident by Theorem 7.7.

)

o.for A * A, (* =

>m r

ril^Avl

In order that the closed linear manifold gene-

necessary and sufficient that

For

-

*:rl

rated by an orthonormal system

Proof .

flan

II

*

The uniqueness of such ot x

uch that

1

1

JE. t.f

(A/1)is

^T"

of x

com-

C A is con-

X6/^

vergent.

Conversely, let sequence.

0.1,

=

2L

^i/, x C A

(/ = 1, 2,...)

Since we have by Theorems 7.7 and 7.8

be a Cauchy

Orthonormal Systems

8)

7,

we conclude that

oC k>x

Furthermore, putting ,11.

a

ia convergent for every A

1, 2,...)

*

I*.

a,

I

(*

<* x

^i^

x

*<*,

Jfe^

29

>

I**

/*,,,

Therefore, if the condition stated in Theorem we can put

=

a.

T"

a

H 0.,,-

C>

c^ A

*

II*

A

.

"* see easily by Theorem 7*7

^H*l*.

7*10 is satisfied, then

and we have by Theorems 7*7 and 7*0

,

^

!<*,* - <**

|*

(j,

B

i. 2,...).

On the other hand we have lift

a.

il

/*-*o Thus we conclude

-a^ H 1 m r 11

lin^

x^A - a

(X^

-oul*

\o(

a

II

manifold generated by

Cx

(

A

*

/I

because

(7,

is a Cauchy sequence by assumption*

(i/.i, 2,...). ft^ (f

&

1, 2,...)

Therefore the closed linear is complete*

)

Recalling Theorem 2-7, we conclude immediately from Theorem 7*10

tem

Theorem 7.11.

In a Hilbert space

a x *R(x*/l)

2L.

8 Let ^

A =

X

K

i

)

for any orthogonal sys-

JB convergent, if and only if

^-x.

XM

ft ,

Dimensions of Hilbert Spaces

be a Hilbert space.

An orthonormal system

aid to be complete, if

(x

,

tx)

CA

for all x

(

X

R^ (

implies

/\

We see easily by Maximal Theorem that there exists a com-

.

plete orthonormal system for every Hilbert space.

Theorem 8.1.

In order that an orthonormal system

be complete, it is necessary and sufficient that

amental in

A

Let

Since

).

such that

A

)

lICLil

5

=

A

A

)

is fund-

)

/\

C^

A.

(

does not coincide

&, then we can find ty Theorem 7.2 an element a 1,

(a,

C>

)

is not complete by definition.

for all

A

be the closed linear space generated by

is complete by Theorem 2*7, if $

with the whole space p^

( -X

\

(

R.

Proof. d

CA

CA

implies

a

for all X If $

/I

R

,

,

and hence

then (a.

,

C*

(

CA )

^ ^y Theorem 7.1.

By virtue of Theorems 8.1 and 7*9, we obtain immediately Theorem 8.2.

For a complete orthonormal system

C* (X

A

)

A

HIL3ERT SPACES

30

of a Hllbert apace

fora

<X

=

&,

o< x

^L

a

e

ft.

may be represented uniquely in the

Theorem 8.). ay sterna in

every

(Chapter I

p,

all complete orthonormal

,

have the same density.

ft

Firstly we consider the case where

Proof.

R

has a complete

orthonormal system compsed only of a finite number of elements.

-

R( V

C,

2,..., K) be a complete orthonormal system in

*>

shall prove by induction that every

X. -f

x

2,...,

79

we can find by Theorem

l)

-fr

K =

We

We as-

1.

For a system

1.

>c

R.

elements are not linearly

In fact, it is evident in the case:

independent.

sume that our assertion is valid for (f*=z 1,

1

Let

^M,*

a

R,

such that

x (/K= 1, 2,...,

C,,

We can suppose obviously

o<

x^

K

4

^s

x-H).

0, without losing the generality.

Then we have

K Thus we can find by assumption

such that

(/*=!, 2,..., x)

i)

Therefore

a^

=

M K

1, 2,*.

(yu

^-

=

,

l) is not linearly independent.

Since every orthonormal system is linearly independent, we conclude hence that every complete orthonormal system is composed of

Next we consider the case where

K

elec^.l-o.

has a complete orthonormal sys-

8,

Then every complete orthonormal

tem composed of infinite elements.

system is composed of infinite elements, as proved just above* <X X

(

x

A

)

and

*

r U=

fr

*

f)

two complete orthonormal system In

b

R

we can find by Theorem 7*6 at most countable elements

For each X 6 A

r^

4^

Let

1,

2,...)

such that

Furthermore, for any

1,

,...).

>

A 8uch that

(

4f

,

ax )

4t

U Jf

.

<*.*.*,... Thus, denoting by

4*^

and

^

r

,

P

aj

=

for

r 4 ^,,

<' "

we can find by definition

Therefore we have

=r

*

cardinal numbers of

A

and

f

respectively,

*

Dimensions of Hilbert Spaces

8)

^<

we have hence

$ >n =

Consequently we have

**v

=

We also can prove likewise **<>*.

>H.

-frt*.

For a Hilbert space R

all complete orthonormal systems in

,

the same density) as proved just above.

3

if and only if the dimension of

X

is isometric to a Hilbert space

ft

ft

If R is isometric to

Proof. (

A Hilbert space

Theorem 8.4-

,

coincides with that of $

also obtain a complete orthonormal system

for the dimension of

ft

Tc A

coincides with that of S

Conversely, we assume that the dimension of

R

,

and let

and

^ >s

x

(

cs>. Ci A

79

r-.

R

$

,

and hence

A Hilbert space ence

a^

6

R

(^

T

coincides with that

ft

o^

(A

(

A

)

ft

,

R

in

ve can find

%

such that

a

such

6

is an isometric transformation from

is isometric

ft,

There-

Therefore, putting

/,\.

V9 see easily by Theorem 7.7 that to

9

).

be complete orthonormal systems

/I )

uniquely complex numbers

2Ho< A

$ we

.

and then we obtain by Theorem 7.10 uniquely J

,

A 6A

that

4

A

$ (A

Corresponding to every x

S respectively.

by Theorem

H

ci>.

and

ft(A/t)

CA

*

.

Tx

by a transformation

ft), then for a complete orthonormal system

of S

have

ft

This same density is called

R.

whe dimension of

.

31

to

.S

by

T

is said to be separable, if we can find a sequ-

which is dense in

1, 2,,..)

R.

Kith this defi-

nition) we have

In order that a Hilbert space

Theorem 8.5.

is necessary and sufficient that tae dimension of'

If

Proof* is dense in

R,

R

then

is separable and a sequence

a^(^

* 1> 2,,..)

ft.

ft

be separable, it

is at most countable

ap

R(k =

2,..)

1,

is obviously fundamental in

R*

By virtue of Theorem 7*?, we can find an erthonormal system composed of at most countable elements C v is A linear combination from

2,...)

Cv

R (*/

is obviously fundamental in

sal system in

R

.

(^

*

=*

1> 2)..*)

l t 2,...).

R,

a^

such that every

Then

,,(*/

~

1

and hence a complete orthonor-

HILBSRT SPACES

32

(Chapter I

Conversely, if we can find a system of at moat countable C,, >|

1, 2,..*)

R. (i/

fi

laments

than, denoting by

,

the totality of linear combinations with coefficient* of rational

ooaplex number* from table and dense In

R

which ia fundamental in

C v ("

R

*& aee easily that

1, 2,,..)

ia coun-

/4

Since every complete orthonormal system in

R

ia obviously fundamental in

we obtain therefore our assertion.

,

Furthermore we also have proved -

6

If a Hllbert apace

R

has a fundamental system

composed of at moat countable elements, than

R

dimension of

R

putting

Weak Convergence

be an inner product apace.

%, (x)

functional on

(A,

R

,

,

as stated in

A sequence

3.

av

(

R

is weakly convergent to

1, 2,...)

that ia, if

a

,

(a ,x

x)

R

a,,

(v

1, 2,...)

and we write

-*,

w-ij^a,,

V^ (^ *

CL f R,

R), we obtain a bounded anti-linear

(x

x)

Corresponding to every

ia said to be weakly convergent to a limit

if

is separable and, the

is at moat countable,

9

Let

R

^

for every

)

as defined in

,

R

x

lig^ With this definition, we see at once by Theorem

?!

that the limit

of a weakly convergent sequence ia determined uniquely.

Furthermore

we hare obviously

w-lim

9*1*

w^lim

(o(. flu u -f

for every complex numbers

^ a^ =

w-lim

implies

Since

^a

||

Theorem 9.2.

=

flafl

c*

,

(3-O ^

w-lim 4^ =

=

t* <x

w-lim

.

by

v

a

.

implies

^^

au

=

c^

,

lim

o/w

CL V

=

A

impliea 9

lim t-

n

-4<*

lim

cx.,11

a* =

^

(X

II

ct

II

.

implies

Therefore usual convergence may be often called

strong convergence against weak convergence* Theorem 9.3-

=

5(2), we obtain by Theorem 3*4

We have obviously by Theorem 3.3 that

A

-f

^

o^a*

"I1!* "^ *"

* &,

ttfc,

For a fundamental manifold

/(

o

3,

.

R,

j

Weak Convergence

9)

55

^

x*A

for every

t,x)

<+<*>, w-llm a*, 3 a,.

then we have

By virtue of Theorem 3*6, we conclude from our aesu*p-

Proof*

R

such

we have by assumption

that

tion that there exist* a bounded anti-linear functional

v-lim

that
(x)

*fa,v

and both Cf and

eR

x

v-lin

f a ^,.

a^sa,

w-lim

Theorem 9>5>

Proof*

-+eo

I

w-djm^

J

Cl H

the formula

have by

Therefore we

Ha^|<

sup

^

oo

-f

,

1

w-Ui^a^aa, li^l

liij **

a,

(a v 4

CL

iapliea

fl

a,,|l

CJt|

(^^

v~ 1

^"

lim "****

**

aB

,

t) =

2a

and

Since we have by

by Theorem 9*2*

4(5)*

1 1 Ha,- a* -adalSla,,! ) Ua > I | a I by a**uaptlon,

l^^^n

Tj* Therefore we have

1

-4a*-

=

a

a^-a| lia

(Xv

JtJ^ji^+ a.*tS

Ve shall consider in the sequel the case where

Recalling Theore

Theorea 9.6.

R

0.

is a Hilbert

5*2, we have obviously

In a Hilbert

ergent, then

1

eonolud*

o-i,

space*

s

* A.

lim^a^

and

(*)

9*

/t

lim

obtain by assunption

fl

fundamental in

i

>4

Since

Proof*

2

,

on

a. t,

hence

Since

.

that is,

>

a^

Theorem 9>4.

have

A

*.

are continuous , we conclude hence that

for every

definition

^

For such

for every

fc(x)

^(a;)

s

M

|

paof

R i

CX|,

,

a* fl< + *o

R.

.

Therefore we conclude by Theorem 9*4 9.7.

In a Hilbert space

ft

,

w-lim *V

*^*O

a^ *

Ct

,

lia IT

*+9&

4*

HILBERT SPACES

54

For a fundamental manifold

Theorem 9.8.

R

i

>

(^ = 1|2,...)

(fr*9*)

Ha* H< +00

sup

**l

,

then

a,

(i/

=

Recalling Theorem

Proof.

(Chapter I

A

lim (,

5.4 0,*,

f^ is

CL (i/

R.

=

(a,,

-{KO

,X

36

on

q>

R such

x

for every

K.

that

.

assumption, we can find then by Theorem

\yy

for every

is weakly convergent

I* 2,...)

A and

we can conclude from assumption

((X,x) =
such that

t

is weakly convergent.

1, 2,...)

= ?(*)

)

a Hilbert space

x

is convergent for every

that there exists a bounded anti-linear functional

Since

of a Hilbert space

x* R

Therefor

.

definition.

"by

Recalling Theorem y.2, we obtain by Theorem 9.8 In a Hilbert space

Theorem 9.9*

is convergent for every

z

t

R

>

then

ft

if (cu ,x

>

(^ = 1,2,...)

)

is

1> 2 f ...)

o.^ (^

weakly cor^

vergeat.

2,...) is convergent for every

then CL V (^ =

l

Proof.

ak R

(f

p =1,

5

Denoting by

= 1, 2 ,...), we see by Theorem 2*7 that

is complete by assumption.

4

and

2,...

a",

,&**}

("=l

(a^.O.^) sup

1)

a^|| <- oo

&

Therefore, putting /) 4*

/4-{x:

-xJ-B},

^8 fundamental, and hence

B

also is fundamental.

Since

(a^Oy*)

=

(*/

2,...) is convergent for every y* =1, 2,..,, and every % k A and i> = 1, 2,..., we conclude by Theorem 9.8 that

In a Hilbert space

Theorem 9.11. (i/

*

1| 2,...)

Proof.

Putting

K^tV^I

Then

cy)

<^^

(f

s a^p (p

K

= ^sup

H a,

^ H^IHI^/uH ^

1, 2;...) 1,

2,...)

ft

if

,

sup

II

d^

It

<

*

co

t

contains a weakly convergent subsequence.

and hence we can find a subsequence (<Xi/ f

!,

for

is weakly convergent.

(k=l,2,...)

Q, U

3

is complete, because

((X^, x) =

then

f

the closed linear manifold generated by

we see by Theorem 7-2 that the union {

if

t

is weakly convergent.

2,...)

t

R

In a Hilbert apace

Theorem 9.10.

il

IT*

,

we have by

for every

a^(f=l,2

>

...)

is convergent for every yu

4(2)

^

,

^

8

such

1, 2 t ...

is weakly convergent by Theorem 9.10.

Recalling Theorem 5.2, we conclude easily

2,...,

1,

that .

Weak Convergence

10)

9,

Theorem 9.12.

A

(A

i

)

suj)

In a Hilbert apace |

<

(a A ,x)| sup AC A

Theorem 9.13.

I

It

+ oo

ax

w-lim^

elements

then

>

lira a.* ->ao

&

=

<*>

-f

a^ = a

are contained in a linear manifold

ft

for a system

.

x

for every

<

II

35 o,x *

6 R, then we have

.

g p (ys

and all

.

1,

2...)

generated by a finite number

;vf

R

of

.

*

'By assumption, we can find an orthonormal system

Proof .

such that every

(/*- 1, 2,..., K) the form

X

= JEZ o/^ CM-

.

6M

nay be repeented uniquely in

^

Putting

'

^.l

X

CM 6/

/!

<*

-

C^ (^=

1, 2 t

'

'

we obtain by assumption for every /* = 1, 2,...,

..)

and hence

Thus we have

lim l

10

Let

T

R

Operators

For a manifold D of

be a linear space .

R

T*

X

P

Here D is called the domain of

T,

into

of D

An operator

:

T

in

is a linear manifold of

X

for every

,

*

D

K,

R R

(

^

and

with domain

-(

D

Tx

:

X^

Df the range of

is said to be linear, if

Now let

D

and complex numbers

R

and

implies

TO

o<

= lin^ Ta. k

^lim

au

Ta

tor

T

.

s

a

,

a

An operator sup

p

nx..f<-jo<,x <

T. D

With this defini-

.

,

T

An operator

be a normed space.

is said to be bounded, if

number

.

T

in

R

with domain

R

is a

=

is said to be continuous . if

...),

D

R

R

and

tion, it is obvious that the range of an linear operator

linear manifold of

a mapping

,

), is said to be an operator in

*o

HTx!l<+

>

T

in t

in

*

D (^ =

R

l f 2,

***& domain

for every

positive

With this definition, we ace easily that a linear ppex*-

with domain

D

is bounded, if and only if

Thus for a bounded linear operator

T

"1th domain

p we define

the noya

HUBERT SPACES

(Chapter I

Tit to mean

II

=

UTII

(l)

sup

T xfl.

II

With this definition we have obvioualy (2)

r*

(I

T

for every

X

In order that a linear operator

T

$

(I

*

H

x

II

T

t that

If a linear operator

Proof*

i

-~

Then we have

T

l/

=

1, 2,...

D

T

Conversely, if

U

=

&

^li^Ta^

U

lls^

Ta.

T

then we shall say that

S on R

,

If

ot

O,

^lim^o-,,

T

for

1

is bounded.

a,

,

>

P

a^

ex*

(

-0.

T

JH^ite^ IITH

T

II

a^- fltll

coincides with the whole space

TS

(7 t

is continuous by definition.

Fr

R

R

,

two operators

to mean

T(Sx)

(X

R);

(x

R);

to mean

T

Tx

S T, then

Sx

we shall say that

)^x

*R

&

(

and

X

Then |

&)

* II

^

II

=

|

rators on

R

is commutative to

^

is obviously a bounded linear ope-

|

T

and

TSiT^Si^^^T

for every complex number

(3)

HTSH

(4)

HT+

(5)

II4TH * KIITfl.

S||

S

.

For two bounded linear- operators definition that all

T

may be considered itself as an operator on

Every complex number ?

rator on

^

for every complex number oc to mean

TS

R:

T

we define an operator

(T-fS)x = and

IT(-p-Ov)H

is continuous, then

is an operator J2l

TSX

T+ S

but

,

Therefore

If the domain of an operator

and

1, 2,...).

(l/

is bounded, then

^im^Ta^-To.!!

T

ia not bounded,

implies by the formula (2)

1> 2,...)

and hence

T

be continuous.

1, 2,...) such that

(i/

D, ^IJ^ --a.,:*

Therefore, if

.

o

with domain

HTa, 11^

*

*.

.

is bounded,

then we can find by definition (l) d^

ttaytffl,

D

UTII IIS

o<

S on

R,

and we have

II,

VTII + l|l,

,

we see easily by

also are bounded linear ope-

10)

Operators

37

Thus the totality of bounded linear operators on ad space.

R

bounded linear operators on tors

%

R

constitutes a norm-

Therefore we can consider convergence of sequences of

(i/

8

1, 2,...)

bounded linear operator JLirn^

T

on

H

T*

A sequence of bounded linear opera-

.

R

on

is said to be uniformly convergent to a

R

if

t

- Til

0.

With this definition we have obviously by the formula (3)

If two seouences of bounded Ijnwir operators T*

Theorem 10.2.

and S ^ (^ s If 2,...) on

near operators -f and . .

)

are uniformly convergent to bounded li-

R

R

oji

also is uniformly convergent to

*7"<S

For a sequence of operators T^

(*/

Tx T

lirn^

we obtain an operator 1, 2 t

..)

on

R

T

is convergent to

1^

(*

lia

i

then we have

X

*

lim

^**

T, ~r

I'**

Proof.

Putting

R,

1^

lif "^**

l

$^ s Jf

S^ - $

a.

.

T

also

is

R

-7^

and

IT.,1.

aup 11

,

II

T.

IK

^

,

*

TS sup

/|

T^

T^xi ^ rifi^x-Sxi *

"T^

lift^

jif

and hence we obtain by assumption

x

R),

For two sequence of bounded linear operators

l t 2,...) on

*"**

X

denoting

>

also is a bounded linear operator on

(i/ =5

"TV

For a sequence of bounded linear operators

Theorem 10.4.

*,

if

then, putting

nr^x

I1TII*

and

,

Furthermore we have obviously by Theorem 2.2

Theorem 10. 3

T

R,

R

(^ = 1, 2,...) are linear, then its limit

obviously linear.

then

!, 2,...) on

and then we shall say that "TV (^

,

*T Here, if all

=

*

for every

1, 2,...) is convergent

(f

"Ty S^ (f * l f 2,

respectively, then

lii_

II

>

we have for every

TC < R.

+

Tw S^x =^T^x

for every

7^

HILBERT SPACES

38

T

For an operator the manifold

Tx

{

TB

>1

K

,

we denote by

With this notation, we have

\

Let & be the closed linear manifold generated

R

.of

For a

T

bounded linear operator

TA

TA.

nerated by

B

Then we

liave

Therefore

IJn^

TB

a^

Ta^ =

T-/

2,...) such that

- 1,

(f

/4i

because

,

>

T

Jim^

a**

=

^

is bounded by assumption.

.

,

corresponding to every

?*(*) = (a

Let R be

For a bounded linear operator "T CL ^

R

l^iOOl

(x on

<{&,

Tx4

h

ceil

II

putting

,Tx)

we obtain an anti-linear functional

(

R

,

R),

nctl1 l|-r

l|

ft

Thus , putting S

This operator

,S

a i

=

^

^

(

R

^

)

,

)" XM

^

for every

(flL.,Tx)a:(^ ,x)

X

and for every

Thus we can find by Theorem 5.4 uniquely an element

e

R,

%

f

saoh that

R

.

R

we obtain an operator S on

called the ad .joint operator of

T

R.

-

.

and denoted by

Therefore we have

"T*.

(T - x ,^

(6)

)

The adjoint operator

for every Z

for every

X

(7)

=

(*i"T)

T * of T

for every

,

e

and complex numbers

HT*H

=

II

T

o

^

6

R.

i3 bounded and

H

T

,

Furthermore

. (3

T

H

Because we have by the formula (>) for every

T*

x,

is linear, because we have

R, and hence by Theorem 7-1

R. ^ IB bounded and we have

Thus

A

is included in the closed linear manifold generated by

a Hilbert space in the sequel. R.

fA

is included in the closed linear manifold ge-

t

Finally we consider operators on a Hilbert space.

on

.

Since A^ is dense in B ty Theorem 2.5, for any

*2 can find

,

R

on

the linear manifold generated by

Denoting by A\

it is evident that

TA

A

*

is included in the closed linear manifold generated by

Proof*

Jl

A of R

on R and a manifold

:

Theorem 10.5. by a manifold

(Chapter I

"* II

<

II

T

II

.

x

R.

Consequently we have ob~

Operators

10)

39

viously by the definition (6)

=

(T*)*

(8)

T,

and hence we have

H(T*)*H <

Ttt =

T*

H

Therefore we have

as proved Just above.

,

T

II

*

II

T*

||

.

II

Concerning adjoint operators, we have obviously by definition * (o(T +/3S)* = 2 T + p S*, (9)

(TS

(10)

K

X

i

&

In a Hilbert apace

Theorem 10.6. linear operators

S* T*.

=

)*

(

TV,

* *

then we have

)

/I

sup

x

t

Tx

||

^

R

2P.*

<

II

8U P

Jb

>

A

* *o

by the formula

4(2) for every

sup

I

UX|l-e 1,A

(T/

$

x

,

J

)

TA x

/I

^

for everj

""""~~""~

TA

of

*

(

*

/t

ve have

)>

R.

sup

I

Tx

II

sup

< + oo J|

|

X^

lUll^?,

A

Thus we obtain by Theoreu 9.12

<+

II

.

Tx *

For the adjoint operators*

Proof.

for a system of bounded

,

(

X

-T\ ^

,

)

|

.

T\^xll<**o, and hence

"XII51, Af/\

,|T Al

aup

=

,

Tx*xM<+-.

=

-lV

*$

-upisup^

Recalling Theorem 10.3} ve obtain by this Theorem 10*6 For a sequence of bounded linear operators

Theorem 10. 7 (*

ss

1

,

2,

. . .

on a Hilbert space g

)

is a bounded linear operator on

R

,

,

_if

and

T

%

IJn^

sup

||

T^

II

>

TV

T

then

< +00.

Furthermore we conclude from Theorem 10.4 In a Hilbert space

Theorem 10.8. tors

T;

we have -

3^

.

lim 1--++0

-

=

(

1, 2,...), If

"-^iga

Proof.

Hm^^-T,

T

a,,

Ta

= S

.

then

x)a

X

T

>

v-lim A'

>oo

d^ =

T * of

'T

,

if

a^ s

w-lim

R.

lim ~

a,

.

For the adjoint opera tor

(To,,, f

lim_..

For a bounded linear operator~

then we have for every

lim

for bounded linear opera-

TS.

T; S k =

Theorem 10.9implies

,

*

(a,

,

T*x)

(a T*x) = ,

(T*

>

X),

flL i

HILBERT SPACES

40 w-lim "+

and tone*

An operator

T &u* - T&* T on a Hilbert

(Tx,jf

space

R,

(Chapter I

la said to be symmetric, if

for every

(x,Tf)

)

x,^**,

With this definition, we have

Theorem 10.10.

R

Every symmetric operator

T*

is a bounded linear operator and

=

T

on a Hilbert apace

T. z

Since we have obvioualy by aasumption for every

Proof*

* R.

^(f,Tt.) we obtain by Theorem 7.1

for every

x

the formula

R

,

and complex number*

Since we have by

.

^

^

Theorem 10.11.

=

(Tx ,x)

and hence

|/T*-H< + oo,

eup

nXS

T

is bo-

|

It is obvious by definition that

satisfies

,

4(2)

we obtain by Theorem 9.12 unded.

<*

T*

If a linear operator

T

(x,Tx)

*

for every

=

T

.

on a Hilbert suace *

/g

.

then

T is

a sym-

metric operator on R.. We have by assumption for every complex number

Proof.

(TX,X) Since

T

n

(

is linear by assumption, we obtain then

X(Tj Thus, putting

,*) t A

\ =

(Tx,^)

*

A(^ ,Tx)+

1, we have

(Tj ,x)^ (Tx,j) and, putting

x =

t*

i

*

obtain

(Tj ,x)- (Tx,f X) (T

Therefore we conclude

,

jj

) (

J

,

Tx)

for every

f{

\

41

CHAPTER II THEORY

SPECTRAL

Projection Operators

11

Let R fold

be a Hilbert space in the sequel.

A of R ,

complement of

the manifold {

x

x

:

A

"*"

A and

For an arbitrary mani-

M

called the orthogonal

Vith th* 8 definition, we have

denoted by A

obviously

R

(1)

A

= io\,

lof=

(2)

A :>B

(3)

AC

(4)

A A* C.\0\.

Since

e.

8

,

A**,

Ai A C X

X* ?

ty (5), ve obtain

A

the other hand we havo

X*

(5)

^

implies

ft,

A

C

/4

"t A

J

(/4

On

ty (2).

)

Therefore we con:lud

(A*)*'*' *V (?)

=

Furthermore we see easily by definition for a system

A)

(*

7, we see easily

Recalling the relations (2), (3), (5) in

A

The orthogonal complement

Theorem 11.1.

A^

* is a closed linear

A

manifold for every manifold

We have further Theorem 11*2.'

A

fif**

for every manifold

i

ty (3), we have (2), (5).

6

BC/j

AX

P

B^^A^^^ = A* B^C/Tby (2).

& ?A

implies

and consequently

**

* A*'*.

A by We

Furiitersnore

is complete by Theorem 2.9, because /^ is complete by assumption.

Thus for any

4

,

A

by definition, and hence

On the other hand,

have therefore ft* =>f*

B

.

be the closed linear manifold generated by > closed linear manifold by Theorem 11.1 and /4*

Let

Proof.

Since A**'

A

is the closed linear manifold generated by

&

a.

B

*

and^=da4ac

^ ind bY Theorem 7-2 ^ for. some

X

I

B

^

R.

such

tfjat

complex number oC

i|{ll .

a

1,

Hert

SPECTRAL THEORY

42 we have

oC

4

we have

<x

*=

-X + -JT

state that for any

X

*

=

6

1C

a

.-X

>

B

* ft

implies

#

*

? B by

(4).

&

Therefore we can

<

^^

>

4

a

*

8

and

1 "*"

and

a

such that

&*

'*"''

X

Such Jj..

such that 3 A--

O,

=

*=!*'%

**

R

a.

we can find

* f&

are determined uni-

,

Because, if

quely.

&

then we have

9

X

* *1 +

*+

-

X ~ X,

J

|Ji

8

*

B

x<

,

l'

x"x

and hence

,

,Jf|8'

;

"*"

= r

t

= ^

^i"5

Therefore we can state

by (4)*

If

11. 3

for any

Let any

X+

=

J>

Hence

by (4)} and hence

= A

= 8"

Consequently we have

In this Proof, we have proved that for any 1"

8

we have then

B'

8

6

Thus we conclude V =

by ()).

&

B

J-

6 ft we can find

Especially for

.

because B C &>

44

because

,

(Chapter II

ct

fl.

that (X a

is a closed linear manifold of

we can find uniquely

0- 6 R.

M

^

X

A

&pct

/4

&

be a closed linear manifold of

J

x

f

M

projection of an element 0L to

X

Then we

&

This operator

.

ed the projection operator of a closed linear operator At pi

=s

M

AA

is call-

PM

.

^y Theorem il.2, we have obviously by definition

PM +

(7)

is called the

and denoted by p^ 0u

on

obtain obviously an operator prt

Since

For

M a* jJ^M* such

Tnis uniquely determined element

.

then

ouch that

in the sequel.

we can find by Theorem 11. 3 uniquely

x+

f /4

ft. t

P^

for the projection operator

=

1>

Recalling the formula (l),

of M**"

P^^

we conclude immediately by definition

?^

(8)

M

-1,

P

- 0.

It is evident by definition that

PM

X

= X implies X

definition, because

*

M

X=X+0,

P^ X

f

X

Convai^ely

M for every f M implies K

jc

MM

X^A^,

.

6

R

Thus

= P 5C M

by

Therefore we can

state

Theorem 11. 4

We have

f^

x

X

>

We see at once by the formula (?) that

if and only if

P^,

x =

*f

PmX

X

Therefore we have by Theorero 11.4

X

Af

if and only

if

Projection Operators

11)

Theorem 11.5. Since

PM X

We have

*

M

=

PM P M

*

R

x

if and only if ,

X

PM

Thus every projection operator (9)

X

for every

PM PM X s

=

p^ x

43

M*.

6

we obtain by Theorem 11.5

for is iden potent:

p^

PM/

Furthermore we can state

Theorem 11.6.

Since

The range of p M

PM x

PM ^

M,

*

M*

* {f

M

coincides with

X

*or every

=

for every ( PM . Pfl*}) This relation yields the formula (7) by

X

y

,

,

f

e fl

R

,

,

that ia.

we have

.

We also obtain likewise

s

<*PM>

Therefore we have

(pM x,j)

do)

y)(pM */^)*

(<rpM

Prom this relation we conclude that the projection operator Therefore

symmetric. rem 10*9

1

and hence

(11)

If

PM

P^

X

g

II

|l

X

1 =

X

R.

6

7(4), we obtain by (7)

HP**!!

1

* I\?M+*

1 II

-

This relation yields (ll) at once, and we obtain further by (7) I!(1-

(12)

Furthermore we have

X

M"

"

1

1"

=

M

Theorem 11.7.

||

P^J.

x

II

=0

by Theorem 11.4, if and only if

Therefore we have /!P/<*-ll

=

tf

X

||

implies

p^X = X.

By virtue of the formulas (9), (10), every projection operator is

a symmetric idenpotent operator on

Theorem 11.8.

R

.

Conversely we have

For a symmetric idenpotent operator

can find uniquely a closed linear manifold

ft

sucj^Jbhat

T

T =

on P/^

is

by Theo-

Furthermore we have

for every

f|

Because, recalling the formula ffXfl

R

is a bounded linear operator on

P^

is continuous.

P^

fl

,

SPECTRAL THEORY

44

T

Since

Proof.

%

have for every

10 symmetric and idenpotent by assumption, we

R

J

,

(TX,S-TJ.) X

Therefore | denoting by

X

= (x,

s 0.

Tg-TTw.)

or T, we have for every

the range

Tx

Tx-f (X -Tx),

Since /4c/4

(Chapter II

X

(

R

x-Tx/f*

A

J"l

'by (3), we conclude hence by definition

T

This relation yields by (?)

Tx

-

X

=

=

P^JU

P^A

x

.

.

By virtue of this Theorem 11.8, every symmetric idenpotent operator on p

also is called a projection operator.

Calculus of Projection Operators

12

For two closed linear manifolds or P^

p^|

to mean

N

ll(?) that PM

^

5

and /V of

R

,

we define

p^ is equivalent to P M *

<

1- PM

is equivalent to

For two projection operators p and

Theorem 12. i.

PM ^ R>

With this definition, we have obviously

.

ll(2) that PM

by the formula

hence by

ftc

M

-%

p^

1

~

Q

9

and

P/v*

P

,

^

ft

is equivalent to every one of the following conditions: 1)

QP

P,

2)

Pft

P,

3)

H

1 fix - pxll

BflTCII

4)

H

Px

H

5)

ftX =

g

H

ftx

MPXII

1

for every

pX

implies

for every

X

R

X

R

.

.

0.

Recalling Theorems 11.4 and 11.6, we see easily by defi-

Proof.

nition that

I!

1 -

p^

implies

<J

QPx

=

px

for every

X R,

that

is,p$Q,

implies the relation l). If

Qp

s p

,

then we have by the formula

lo(lO)

((2p)* = p*CL,* = Pft,

p * p

because every projection operator is symmetric.

Tf

Thus l) implies 2).

p = pft, then we have by the formulas (9), (10) in

(PX and nence by

,

Qx -px)

= (x,

Pax-

Px) = 0,

7(4)

Px

||*

for every

11

Calculus of Projection Opera ton

12) Thufl 2)

45

It la evident that j) implies 4), and that

implies 3)*

Recalling Theorem 11.3 and the formula

implies 5).

P ^

easily by definition that 5) implies

4)

ll(2), we see

Therefore our asser-

fi.

tion is established.

For two manifolds that is,

x x

A

*

If

tion, we have

A< J-

PM *

(

,

(*

Py

Conversely,

pH py

3

Jf

*

then, since f

)

=

With this definition,

PM x

(

* At

,

and hence by

)

a

x

t

If P

PCI *

,

3

11(10) for every

x

*

.

?!

PM Pyj) =

fc

,

<>.

pa

P

=

p

fl,

PQ

is symmetric,

P.

then we have by

is symmetric .

(pa)(pa)

,

10(10)

(pft)*= a*p* = Q.

Q

$ P

is a projection operator, then

ft

(PL*,j>) *(^,flP^) Q,

X

For two projection operators p and

and hence by the formula

and hence P

* >V *V defini-

11(10)

also is a projection operator if and only if p(J

Conversely if

#

P>v

for every

implies by

s

we

,

by Theorem 11.6.

Theorem 12.).

yroof .

A C &\

to mean

.

by Theorem

Py?>

(fVt*

y

A JL 8

we define

,

and

Py

P/* P//#

,

p^ p^

a.

PM

^>

Thus we conclude

^

A

&

B implies

if and only if

Proof.

and hence

x

of

For two closed linear manifolds ^f and /V

Theorem 12.2.

M J. y

B

and

for every

it is evident that

have

A

ll(lO)

(*, PflJ),

Furthermore we have

(Qp)(pa)=apa

that is, P Ci is idenpotent.

Therefore

=

pQft

=

pQ,

pfl is a projection opera-

tor by Theorem 11.8.

For two projection operators p and ft,

Theorem 12.4.

also is a projection operator if and only if P Proof.

If

potent, and hence

p+

ft

Q.

p

*

ft

~

is a projection operator, then

p+

ft

is iden-

SPECTRAL THEORY

46

(Chapter II

This relation yields

= 0,

OP

PQ, +

P41+

QP&+QPQ~0*

and further

consequently PQ, = Conversely if

Pft = 0,

fi

Thus we obtain

p& &

ft

and

,

.

PQ

=

then we have

,

ft

P =

+

ft

by Theorem 12.2,

and hence

that is, P +

P

-f

Q,

-p

is symmetric by

10(9),

is a projection operator by Theorem 11.8.

is a projection operator if and only if ^ia^MfcirfliaMiiii^aii^iiiiiapiiii^iiM >~uMm*~~m

Pi/ /IT

I

*

PM Proof.

If

#

For two closed linear manifolds

Theorem 12.5*

?^ -

Since

is idenpotent.

Q,

PM - P^

=

P^

I

I

and tf of

ft

> R/, and then p^ '/V /H * '

*

'

'

'

PAfA**

is a projection operator, then we have by

Theorem 12*4

because

PM *

and hence

"

(

P>y

PM

*"

^ ^ PM ^ Theorem 12.1*

Conversely if P^

< PM

,

= Pv PM =

then P^ P^

P,v

^y Theorem 12.1,

and consequently

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