# proof of classification of separable Hilbert spaces

Since $H$ is separable, there exists a countable  dense subset $S$ of $H$. Choose an enumeration of the elements of $S$ as $s_{0},s_{1},s_{2},\ldots$. By the Gram-Schmidt orthonormalization procedure, one can exhibit an orthonormal set  $e_{0},e_{1},e_{2},\ldots$ 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

 $\|v-s_{m_{n}}\|\leq 2^{-n}$

The sequence $(s_{m_{0}},s_{m_{1}},s_{m_{2}},\ldots)$ is a Cauchy sequence  because

 $\|s_{m_{i}}-s_{m_{j}}\|\leq\|s_{m_{i}}-v\|+\|v-s_{m_{j}}\|\leq 2^{-i}+2^{-j}$

Hence the limit of this sequence must lie in the Hilbert space spanned by $\{s_{0},s_{1},s_{2},\ldots\}$, which is the same as the Hilbert space spanned by $\{e_{0},e_{1},e_{2},\ldots\}$. Thus, $\{e_{0},e_{1},e_{2},\ldots\}$ is an orthonormal basis  for $H$.

To any $v\in H$ associate the sequence $U(v)=(\langle v,s_{0}\rangle,\langle v,s_{1}\rangle,\langle v,s_{2}\rangle,\ldots)$. That this sequence lies in $\ell^{2}$ follows from the generalized Parseval equality

 $\|v\|^{2}=\sum_{k=0}^{\infty}\langle v,s_{k}\rangle$

which also shows that $\|U(v)\|_{\ell^{2}}=\|v\|_{H}$. On the other hand, let $(w_{0},w_{1},w_{2},\ldots)$ be an element of $\ell^{2}$. Then, by definition, the sequence of partial sums $(w_{0}^{2},w_{0}^{2}+w_{1}^{2},w_{0}^{2}+w_{1}^{2}+w_{2}^{2},\ldots)$ is a Cauchy sequence. Since

 $\|\sum_{i=0}^{m}w_{i}e_{i}-\sum_{i=0}^{n}w_{i}e_{i}\|^{2}=\sum_{i=0}^{m}w_{i}^% {2}-\sum_{i=0}^{n}w_{i}^{2}$

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

Title proof of classification of separable Hilbert spaces ProofOfClassificationOfSeparableHilbertSpaces 2013-03-22 14:34:11 2013-03-22 14:34:11 rspuzio (6075) rspuzio (6075) 5 rspuzio (6075) Proof msc 46C15 VonNeumannAlgebra