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

[quasar987]

Homework Statement

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.

Homework 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

The Attempt at a Solution

 

[lippyka]

I am confused about proving that E is a maximal orthonormal set. It seems like this would require showing that any new function we add to the set, which is orthogonal to all of the elements in the set, must be equal to 0 almost everywhere. Is there a way to show this? I am also not sure how to use (2) or (3) to prove E is a basis. For (2), I'm not sure how to show that the orthogonal complement of E is trivial without knowing something about the Fourier series expansion. For (3), I'm not sure how to show that the span of E is dense without using the fact that L² functions converge to their Fourier series.