(adsbygoogle = window.adsbygoogle || []).push({}); 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]

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# Closed subset of vectorspace

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