Isomorphic Hilbert spaces

As part of a broader proof , I need to show that every two separable Hilbert spaces (that contains a dense countable set) are isomorphic (the linear mapping from one space to the other is injective and isometric if I say right).

I'd be happy to get any help on this.

Solution 1:

It is true if we assume the space to be infinite dimensional. Consider $H_1$ and $H_2$ two infinite dimensional separable Hilbert spaces, say over $\Bbb C$. If $(d_n,n\in\Bbb N)$ is a countable dense subset of $H_1$, using Gramm-Schmidt orthonormalization, we can find a Hilbert basis $(v_n,n\in\Bbb N)$ of $H_1$, that is, an orthonormal family, whose span is dense in $H_1$. Similarily, let $(w_n,n\in\Bbb N)$ a Hilbert basis of $H_2$. We define $$T\colon H_1\to H_2,\quad T\left(\sum_{j=0}^{+\infty}a_jv_j\right)=\sum_{j=0}^{+\infty}a_jw_j,$$ where $(a_j,j\in\Bbb N)$ is a sequence of real numbers, whose all but finitely many terms are $0$. This is actually defined only on the linear span of the sequence $(v_j,j\in\Bbb N)$, but we can see that $T$ is an isometry, hence it can be extended to the whole space, being still an isometry.