proof of classification of separable Hilbert spaces

The strategy will be to show that any separablePlanetmathPlanetmath, infinite dimensional Hilbert spaceMathworldPlanetmath H is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to 2, where 2 is the space of all square summable sequences. Then it will follow that any two separable, infinite dimensional Hilbert spaces, being equivalent to the same space, are equivalent to each other.

Since H is separable, there exists a countableMathworldPlanetmath dense subset S of H. Choose an enumeration of the elements of S as s0,s1,s2,. By the Gram-Schmidt orthonormalization procedure, one can exhibit an orthonormal setMathworldPlanetmath e0,e1,e2, such that each ei is a finite linear combinationMathworldPlanetmath of the si’s.

Next, we will demonstrate that Hilbert space spanned by the ei’s is in fact the whole space H. Let v be any element of H. Since S is dense in H, for every integer n, there exists an integer mn such that


The sequence (sm0,sm1,sm2,) is a Cauchy sequencePlanetmathPlanetmath because


Hence the limit of this sequence must lie in the Hilbert space spanned by {s0,s1,s2,}, which is the same as the Hilbert space spanned by {e0,e1,e2,}. Thus, {e0,e1,e2,} is an orthonormal basisMathworldPlanetmath for H.

To any vH associate the sequence U(v)=(v,s0,v,s1,v,s2,). That this sequence lies in 2 follows from the generalized Parseval equality


which also shows that U(v)2=vH. On the other hand, let (w0,w1,w2,) be an element of 2. Then, by definition, the sequence of partial sums (w02,w02+w12,w02+w12+w22,) is a Cauchy sequence. Since


if m>n, the sequence of partial sums of k=0wiei is also a Cauchy sequence, so k=0wiei convergesPlanetmathPlanetmath and its limit lies in H. Hence the operator U is invertiblePlanetmathPlanetmathPlanetmath and is an isometry between H and 2.

Title proof of classification of separable Hilbert spaces
Canonical name ProofOfClassificationOfSeparableHilbertSpaces
Date of creation 2013-03-22 14:34:11
Last modified on 2013-03-22 14:34:11
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 5
Author rspuzio (6075)
Entry type Proof
Classification msc 46C15
Related topic VonNeumannAlgebra