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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 thatwe 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

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

Can you offer guidance or do you also need help?

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