PlanetMath: classification of Hilbert spaces

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classification of Hilbert spaces

(Theorem)

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 and are isometrically isomorphic if and only if they have the same dimension, i.e. if and only if an orthonormal basis of has the same cardinality of an orthonormal basis of .

This will be achieved by proving that every Hilbert space is isometrically isomorphic to an $\ell^2(X)$ space, where has the cardinality of any orthonormal basis of the Hilbert space in consideration.

$\,$

Theorem 1 - Suppose is an Hilbert space and let be a set that indexes one (and hence, any) orthonormal basis of . Then, is isometrically isomorphic to $\ell^2(I)$ .

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Theorem [Classification of Hilbert spaces] - Two Hilbert spaces and are isometrically isomorphic if and only if they have the same dimension.

$\,$

Proof of Theorem 1: Let $\{e_i\}_{i \in I}$ an orthonormal basis indexed by the set . Let $U: H \longrightarrow \ell^2(I)$ be defined by

$\displaystyle Ux \,(i) := \langle x, e_i \rangle$

We claim that

is an isometric isomorphism. It is clear that

is linear. Using Parseval's equality and the definition of norm in $\ell^2(I)$ it follows that

$\displaystyle \Vert x\Vert^2 = \sum_{i \in I} \vert\langle x, e_i \rangle\vert^2 = \sum_{i \in I} \vert Ux\,(i)\vert^2 = \Vert Ux\Vert^2_{\ell^2(I)}$

We conclude that

is isometric. It remains to see that it is surjective (since injectivity follows from the isometric condition).

Let $f \in \ell^2(I)$ . By definition of the space $\ell^2(I)$ we must have $\displaystyle \sum_{i \in I} \vert f(i)\vert^2 < \infty$ , and therefore, using the general version of the Riesz-Fischer theorem, the series $\displaystyle \sum_{i\in I} f(i)e_i$ converges to an element $x_0 \in H$ . We now see that

$\displaystyle Ux_0 \,(j) = \langle x_0, e_j \rangle = \sum_{i \in I} f(i)\langle e_i, e_j\rangle = f(j)$

or in other words,

. Hence,

is surjective. $\square$

$\,$

Proof of the classification theorem :

$(\Longrightarrow)$ Of course, if the Hilbert spaces and are isometrically isomorphic, with isometric isomorphism , then if $\{e_i\}_{i \in I}$ is an orthonormal basis for than $\{Ue_i\}_{i \in I}$ is an orthonormal basis for . Hence, and have the same dimension.

$(\Longleftarrow)$ If the Hilbert spaces and have the same dimension, then we can index any orthonormal basis of and any orthonormal basis of by the same set . Using Theorem 1 we see that and are both isometrically isomorphic to $\ell^2(I)$ . Hence and are isometrically isomorphic. $\square$

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See Also: $\ell^p(X)$ space, orthonormal basis, classification of separable Hilbert spaces

Other names:

Hilbert spaces of the same dimension are isometrically isomorphic

Also defines:

every Hilbert space is isometrically isomorphic to a $\ell^2(X)$ space

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Cross-references: converges, series, Riesz-Fischer theorem, surjective, isometric, norm, Parseval's equality, clear, indexed by, indexes, orthonormal basis, isometrically isomorphic, cardinality, bases, orthonormal, well defined, dimension of a Hilbert space, inner product, structure, vector space, preserves, unitary transformation, dimension, isometric isomorphism, Hilbert spaces

This is version 6 of classification of Hilbert spaces, born on 2008-03-22, modified 2008-03-22.
Object id is 10435, canonical name is ClassificationOfHilbertSpaces.
Accessed 278 times total.

Classification:

AMS MSC:	46C05 (Functional analysis :: Inner product spaces and their generalizations, Hilbert spaces :: Hilbert and pre-Hilbert spaces: geometry and topology )
	46C15 (Functional analysis :: Inner product spaces and their generalizations, Hilbert spaces :: Characterizations of Hilbert spaces)

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