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Closed subset of vectorspace

  1. Feb 3, 2012 #1
    1. The problem statement, all variables and given/known data
    W is a subset of C[-Pi,Pi] consisting of all finite linear combinations:
    1,cos(nx),sin(nx)
    i) Show that W is a subspace of C[-Pi,Pi]
    ii) Is W closed in C[-Pi,Pi]. Hint from Fourier analysis: For x in [-Pi,Pi]:
    [itex]|x^2-(\dfrac{\pi^2}{3}+4\sum\limits_{n=1}^N\dfrac{(-1)^{n}\cos(nx)}{n^2})|\leq 4\sum\limits_{n=N+1}^{\infty} \dfrac{1}{n^2}[/itex]
    2. Relevant equations
    C[-Pi,Pi] could be equipped with a norm
    Lemma: W is closed <=> For any convergent sequence [itex]\{ v_k \}_{k=1}^\infty[/itex] of elements in W the V = limit (vk) for k->infinity also belongs to W.

    3. The attempt at a solution
    I have shown that W is a subspace by realising that linearcombinations of sines cosines and 1's are also in C[-Pi,Pi].
    ii)
    I concluded as sum(1/n^2) is convergent and the left hand side is less than or equal(also convergent) to for x in [-Pi,Pi], then V is closed in C[-Pi,Pi]
     
  2. jcsd
  3. Feb 3, 2012 #2

    lanedance

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    Homework Helper

    1) looks reasonable

    2) I'm not too sure I understand your argument or the hint...

    As a thought exercise how about taking your sequence of functions as the Fourier approximations to the heaviside function
     
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