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

  1. Jan 24, 2008 #1

    quasar987

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    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
     
  2. jcsd
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