How to show Fourier series development for even function

Question

With $\displaystyle c_n : t \mapsto \sqrt{\frac{2}{\pi}}\cos\left(nt\right)$ i've shown that $\underset{n \in \mathbb{N}}{\text{Span}}\left(c_n\right)$ was dense in the set $G$ of real continuous functions on $\left[0,\pi\right]$. ( not necessarily $2 \pi$ periodic ). And i've a theorem that says that for $x \in G$ the orthogonal projection $p_{F_n}$ on $F_n=\text{Span}\left(c_0,c_1,\dots,c_n\right)$ satisfies $$ p_{F_n}\left(x\right) \underset{n \rightarrow +\infty}{\rightarrow}x $$ Am i wrong ? How can I use this to show that all function $2\pi$-periodic even continuous function on $\left[0,\pi\right]$ satisfies $$ f=\sum_{n=0}^{+\infty}\langle f,c_n\rangle c_n $$ with $\displaystyle \langle f,g \rangle =\int_{0}^{\pi}f(t)g(t)\text{d}t$

Answer 1

Your last point $f=\sum_{n=0}^{+\infty}\langle f,c_n\rangle c_n$ for even continuous functions is surely not true at it exist continuous real maps with diverging Fourier series.