isometric embedding of l^2

Question

CLAIM: Let $H$ be an infinite dimensional $\mathbb{R}$-Hilbert space. Then the $\ell^2$ sequence space can be embedded in $H$.

I think it could be true since every Hilbert space has an orthonormal basis $\{x_i\}_{i\in I}$ and so the map

$\Phi:H\rightarrow \ell^2,x\mapsto<x,x_i>_{i\in I}$ is an isometric isomorphism.

Am I missing something?

Answer 1

Choose a sequence $x_{i(n)}$ from your orthonormal basis. An embedding is then $\ell^2 \ni (\alpha_n)_n \mapsto \sum_n \alpha_n x_{i(n)} \in H$. What you write is an ismomorphism $H\to \ell^2(I)$.