classification of Hilbert spaces

classification of Hilbert spaces

Hilbert spaces

can be classified, up to

isometric isomorphism

, according to their

dimension

. Recall that an isometric isomorphism of Hilbert spaces is an

unitary transformation

, therefore it

preserves

the

vector space

structure

along with the

inner product

structure (hence, preserving also the topological structure). Recall also that the dimension of a Hilbert space is a well defined

concept

, i.e. all orthonormal bases of an Hilbert space share the same

cardinality

.

The classification theorem we describe here states that two Hilbert spaces

H

1

and

H

2

are isometrically isomorphic if and only if they have the same dimension, i.e. if and only if an

orthonormal basis

of

H

1

has the same cardinality of an orthonormal basis of

H

2

.

This will be achieved by proving that every Hilbert space is isometrically isomorphic to an

ℓ

2

⁢

(

X

)

space (

http://planetmath.org/EllpXSpace

), where

X

has the cardinality of any orthonormal basis of the Hilbert space in consideration.

Theorem

1 -

Suppose

H

is an Hilbert space and let

I

be a set that indexes one (and hence, any) orthonormal basis of

H

. Then,

H

is isometrically isomorphic to

ℓ

2

⁢

(

I

)

.

Theorem [

Classification of Hilbert spaces

] -

Two Hilbert spaces

H

1

and

H

2

are isometrically isomorphic if and only if they have the same dimension.

Proof of Theorem 1:

Let

{

e

i

}

i

∈

I

an orthonormal basis

indexed by

the set

I

. Let

U

:

H

⟶

ℓ

2

⁢

(

I

)

be defined by

U

⁢

x

⁢

(

i

)

:=

⟨

x

,

e

i

⟩

We claim that

U

is an isometric isomorphism. It is clear that

U

is linear. Using Parseval’s equality and the

definition

of norm in

ℓ

2

⁢

(

I

)

it follows that

∥

x

∥

2

=

∑

i

∈

I

|

⟨

x

,

e

i

⟩

|

2

=

∑

i

∈

I

|

U

⁢

x

⁢

(

i

)

|

2

=

∥

U

⁢

x

∥

ℓ

2

⁢

(

I

)

2

We conclude that

U

is isometric. It remains to see that it is

surjective

(since injectivity follows from the isometric condition).

Let

f

∈

ℓ

2

⁢

(

I

)

. By definition of the space

ℓ

2

⁢

(

I

)

we must have

∑

i

∈

I

|

f

⁢

(

i

)

|

2

<

∞

, and therefore, using the

Riesz-Fischer theorem

, the series

∑

i

∈

I

f

⁢

(

i

)

⁢

e

i

converges

to an

element

x

0

∈

H

. We now see that

U

⁢

x

0

⁢

(

j

)

=

⟨

x

0

,

e

j

⟩

=

∑

i

∈

I

f

⁢

(

i

)

⁢

⟨

e

i

,

e

j

⟩

=

f

⁢

(

j

)

or in other ,

U

⁢

x

0

=

f

. Hence,

U

is surjective.

□

Proof of the classification theorem :

•

(

⟹

)

Of course, if the Hilbert spaces

H

1

and

H

2

are isometrically isomorphic, with isometric isomorphism

U

, then if

{

e

i

}

i

∈

I

is an orthonormal basis for

H

1

than

{

U

⁢

e

i

}

i

∈

I

is an orthonormal basis for

H

2

. Hence,

H

1

and

H

2

have the same dimension.

•

(

⟸

)

If the Hilbert spaces

H

1

and

H

2

have the same dimension, then we can index any orthonormal basis of

H

1

and any orthonormal basis of

H

2

by the same set

I

. Using Theorem 1 we see that

H

1

and

H

2

are both isometrically isomorphic to

ℓ

2

⁢

(

I

)

. Hence

H

1

and

H

2

are isometrically isomorphic.

□

Title

classification of Hilbert spaces

Canonical

name

ClassificationOfHilbertSpaces

Date of creation

2013-03-22 17:56:18

Last modified on

2013-03-22 17:56:18

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

10

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46C15

Classification

msc 46C05

Synonym

Hilbert spaces of the same dimension are isometrically isomorphic

Related topic

EllpXSpace

Related topic

OrthonormalBasis

Related topic

ClassificationOfSeparableHilbertSpaces

Related topic

CategoryOfHilbertSpaces

Related topic

RieszFischerTheorem

Related topic

QuantumGroupsAndVonNeumannAlgebras

Defines

every Hilbert space is isometrically isomorphic to a

ℓ

2

⁢

(

X

)

space

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