Let $V$ denote the subset of $C[-\pi,\pi]$ consisting of all finite linear combinations of functions
$1, \cos x, \cos 2x, ... \cos nx, ..., \sin x, \sin 2x, ... \sin n2, ... $
I want to examine if $C[-\pi,\pi[$ is closed in $V$.
From Fourier analysis it is known that for $x\in[-\pi,\pi]$ that
$\left|x^2 - \left( \frac{\pi}{3} + 4 \sum_{n=1}^N \frac{(-1)^2}{n^2} \cos nx \right) \right| \leq 4 \sum_{n=N+1}^\infty \frac{1}{n^2}$
I want to use this result as a starting point.
My idea is that if I can find a sequence of functions in V that may or may not converge to a function that is (not) in V. Then I can tell whether V is (non)-closed.
Thanks.
I guess u meant to find $V$ is closed in $C[-\pi,\pi]$, if so then you are almost done.. Define $f_n(x) = \frac{\pi}{3}+4\sum_{n=1}^{N}\frac{(1)^2}{n^2}\cos nx$. Then we have $|x^2 - f_n(x)|\rightarrow 0$ for $x\in[-\pi, \pi]$. So we have $x^2 = \frac{\pi}{3}+4\sum_{n=1}^{\infty}\frac{(1)^2}{n^2}\cos nx$ for $x\in[-\pi,\pi]$. So remembering orthogonality of $\cos mx$ and $\sin nx$ for $m\neq n$, we concluded, $x^2\notin V$. But that would imply $V$ is not closed.