Every bounded linear operator $T$ between real Hilbert spaces is $T(x) = \sum \langle x,f_j\rangle\, e_j$ Let $T:H_1 \rightarrow H_2$, where $H_1$ and $H_2$ are real hilbert spaces and $T$ is a bounded linear operator.  Prove the following:
suppose $\{e_j\}$ an orthonormal basis for $H_2$, show that there exist a sequence $\{f_j\}$ in $H_1$ such that for all $x \in H_1$:
$T(x) = \sum \langle x,f_j\rangle_1 e_j$
My guess: 
I know that because $\{e_j\}$ is an orthonormal basis, we get $T(x) = \sum \langle T(x),e_j\rangle_2 e_j$. So we need to find a sequence $\{f_j\}$ such that:
$\langle x,f_j\rangle_1 = \langle T(x),e_j\rangle_2$.
I think i need to use Riesz-Frechet but i dont see how...

For each $j$, the map $x\mapsto \langle Tx,e_j\rangle_2$ is a bounded linear map from $H_1$ to $\mathbb R$, so by Fréchet–Riesz there exists $f_j\in H_1$ such that $\langle Tx,e_j\rangle_2 = \langle x,f_j\rangle_1$.
