Orthogonal representation of finite operator I would like to know if my proof is correct.
Statement: Let $T$ be a finite rank operator on a Hilbert space $\mathscr{H}$. Show that $\forall \, h \, \in \mathscr{H}, \, T(h)$ can be written as $T(h) = \sum_{i=1}^n\langle h, e_i \rangle f_i$ where $\{e_i\}$ is an orthonormal basis of $\mathscr{H}$ 
Attempted Proof: $T(h)$ can be written as $T(h) = \sum_{i=1}^n\langle T(h), e_i \rangle e_i = \sum_{i=1}^n\langle h, T^*(e_i) \rangle e_i$.
Now, if we let $f_i=T^*(e_i)$ then:
$T(h) = \sum_{i=1}^n\langle T(h), e_i \rangle e_i = \sum_{i=1}^n\langle h, T^*(e_i) \rangle e_i = \sum_{i=1}^n\langle h, f_i \rangle e_i$ 
And this is where I get stuck.

I don't see the point of your last computation: you have already obtained what you are looking for before the "And if we take..."
You got your $e_i$ and $f_i$ in the wrong spots, but that's just a labeling issue: you should have started with $\{f_i\}$ as the orthonormal basis. 
On a deeper side, why would $\{T^*e_i\}$ and orthonormal basis? In general, it isn't. 
