# The Hilbert space L²([0,2pi], R) and Fourier series.

1. Jan 24, 2008

### quasar987

1. The problem statement, all variables and given/known data
Is there a way to prove that E={sin(nx), cos(nx): n in N u {0}} is a maximal orthonormal basis for the Hilbert space L²([0,2pi], R) of square integrable functions (actually the equivalence classes "modulo equal almost everywhere" of the square integrable functions)?

I mean, I am asking if we can show directly, using the definition or some other characterization, that E is a Hilbert space basis for L², so that we can conclude that L² functions are equal to their Fourier series. In other words, we can't use the fact that L² functions converge to their Fourier series to show that E is maximal.

2. Relevant equations

Relevant characterizations of "E is a hilbert space basis" that I am aware of:

(1) E is a maximal orthonormal set
(2) the orthogonal complement of E is trivial
(3) the span of E is dense

3. The attempt at a solution