proof of classification of separable Hilbert spaces

The strategy will be to show that any separable, infinite dimensional Hilbert space $H$ is equivalent to ${\mathrm{\ell }}^2$, where ${\mathrm{\ell }}^2$ 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 countable dense subset $S$ of $H$. Choose an enumeration of the elements of $S$ as $s_0,s_1,s_2,\mathrm{\dots }$. By the Gram-Schmidt orthonormalization procedure, one can exhibit an orthonormal set $e_0,e_1,e_2,\mathrm{\dots }$ such that each $e_i$ is a finite linear combination of the $s_i$’s.

Next, we will demonstrate that Hilbert space spanned by the $e_i$’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 $m_n$ such that

$$\parallel v-s_{m_n}\parallel \le 2^{-n}$$

The sequence $(s_{m_0},s_{m_1},s_{m_2},\mathrm{\dots })$ is a Cauchy sequence because

$$\parallel s_{m_i}-s_{m_j}\parallel \le \parallel s_{m_i}-v\parallel +\parallel v-s_{m_j}\parallel \le 2^{-i}+2^{-j}$$

Hence the limit of this sequence must lie in the Hilbert space spanned by $\{s_0,s_1,s_2,\mathrm{\dots }\}$, which is the same as the Hilbert space spanned by $\{e_0,e_1,e_2,\mathrm{\dots }\}$. Thus, $\{e_0,e_1,e_2,\mathrm{\dots }\}$ is an orthonormal basis for $H$.

To any $v\in H$ associate the sequence $U(v)=(⟨v,s_0⟩,⟨v,s_1⟩,⟨v,s_2⟩,\mathrm{\dots })$. That this sequence lies in ${\mathrm{\ell }}^2$ follows from the generalized Parseval equality

$${\parallel v\parallel }^2=\sum _{k=0}^{\mathrm{\infty }}⟨v,s_k⟩$$

which also shows that ${\parallel U(v)\parallel }_{{\mathrm{\ell }}^2}={\parallel v\parallel }_H$. On the other hand, let $(w_0,w_1,w_2,\mathrm{\dots })$ be an element of ${\mathrm{\ell }}^2$. Then, by definition, the sequence of partial sums $(w_0^2,w_0^2+w_1^2,w_0^2+w_1^2+w_2^2,\mathrm{\dots })$ is a Cauchy sequence. Since

$${\parallel \sum _{i=0}^m{w}_i{e}_i-\sum _{i=0}^n{w}_i{e}_i\parallel }^2=\sum _{i=0}^m{w}_i^2-\sum _{i=0}^n{w}_i^2$$

if $m>n$, the sequence of partial sums of ${\sum }_{k=0}^{\mathrm{\infty }}{w}_i{e}_i$ is also a Cauchy sequence, so ${\sum }_{k=0}^{\mathrm{\infty }}{w}_i{e}_i$ converges and its limit lies in $H$. Hence the operator $U$ is invertible and is an isometry between $H$ and ${\mathrm{\ell }}^2$.

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