I am looking at the inner product space $L_{2 \pi}^2$ of square integrable 2$\pi$-periodic functions on $\Re$, with the inner product defined by: $<f,g>$ = $\int_{0}^{2\pi} f(x)g(x) dx$.
For $f$ $\in$ $L_{2 \pi}^2$, let $S_nf$ be the best $L^2$-approximation to $f$ out of $\prod_n$, where $\prod_n$ is the space of all trigonometric polynomials of degree at most $n$.
Since for every $f$ $\in$ $L_{2 \pi}^2$, $||S_nf||_2 \leq ||f||_2$. (Proven with the use of Bessel's inequality)
How do I prove that for every $f$ $\in$ $L_{2 \pi}^2$, $||f-S_nf||_2 \rightarrow 0$ as $n \rightarrow \infty$?
I believe at some point, we can use the concept that $C_{2\pi}$ is dense in ($L_{2 \pi}^2$, $||. ||_2$) where $C_{2\pi}$ is the space of all continuous $2\pi$ periodic functions on $\Re$.
Thank you for your help!