# Closed subset of vectorspace

1. Feb 3, 2012

### dikmikkel

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]:
$|x^2-(\dfrac{\pi^2}{3}+4\sum\limits_{n=1}^N\dfrac{(-1)^{n}\cos(nx)}{n^2})|\leq 4\sum\limits_{n=N+1}^{\infty} \dfrac{1}{n^2}$
2. Relevant equations
C[-Pi,Pi] could be equipped with a norm
Lemma: W is closed <=> For any convergent sequence $\{ v_k \}_{k=1}^\infty$ 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. Feb 3, 2012

### lanedance

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