# Hilbert basis of $L^2([-1,1])$?

Could you please specify hilbert basis of $L^2([-1,1])$? How will be the representation of a function f $\in L^2([-1,1])$ by means of its Fourier series?
My solution:
$E_k=1/\sqrt2 e^{kit\pi}, k\in Z$
$f=\sum_{k \in Z} c_kE_k$
$c_k=<f,E_k> =\int_{-1}^1 f(t) \overline{E_k(t)}dt$

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Do you know what a Fourier series and an hilbert basis are ? –  Damien L Feb 4 '13 at 11:51
Yes, I know. I would like to check that my solution is correct or not? –  Ali Ismayilov Feb 4 '13 at 11:57
"I would like to check my solution is correct or not" What is your solution? –  Willie Wong Feb 4 '13 at 11:58
I am editing now –  Ali Ismayilov Feb 4 '13 at 11:59
Have you checked that the basis elements ($E_k$) are orthogonal to each other and are of unit norm? –  Berci Feb 4 '13 at 12:58