classification of Hilbert spaces

Hilbert spacesMathworldPlanetmath can be classified, up to isometric isomorphism, according to their dimensionPlanetmathPlanetmathPlanetmath. Recall that an isometric isomorphism of Hilbert spaces is an unitary transformation, therefore it preserves the vector spaceMathworldPlanetmath structureMathworldPlanetmath along with the inner productMathworldPlanetmath 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 H1 and H2 are isometrically isomorphic if and only if they have the same dimension, i.e. if and only if an orthonormal basis of H1 has the same cardinality of an orthonormal basis of H2.

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

TheoremMathworldPlanetmath 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 2(I).

Theorem [Classification of Hilbert spaces] - Two Hilbert spaces H1 and H2 are isometrically isomorphic if and only if they have the same dimension.

Proof of Theorem 1: Let {ei}iI an orthonormal basis indexed by the set I. Let U:H2(I) be defined by


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 2(I) it follows that


We conclude that U is isometric. It remains to see that it is surjectivePlanetmathPlanetmath (since injectivity follows from the isometric condition).

Let f2(I). By definition of the space 2(I) we must have iI|f(i)|2<, and therefore, using the Riesz-Fischer theorem, the series iIf(i)ei converges to an element x0H. We now see that


or in other , Ux0=f. Hence, U is surjective.

Proof of the classification theorem :

  • () Of course, if the Hilbert spaces H1 and H2 are isometrically isomorphic, with isometric isomorphism U, then if {ei}iI is an orthonormal basis for H1 than {Uei}iI is an orthonormal basis for H2. Hence, H1 and H2 have the same dimension.

  • () If the Hilbert spaces H1 and H2 have the same dimension, then we can index any orthonormal basis of H1 and any orthonormal basis of H2 by the same set I. Using Theorem 1 we see that H1 and H2 are both isometrically isomorphic to 2(I). Hence H1 and H2 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 2(X) space