How can I deduce that Fourier series of $f$ converges to $f$ in L^2([-L,L])

Question

The question said:

Let $L > 0$ and for $n \in \mathbb{Z}$ we take $$g_{n}(x) = e^{in \pi x/L},$$ We also define the Fourier coefficients of a function $f \in L^{1}([-L,L])$ to be $$c_{n}(f) = \frac{1}{2L} \int_{-L}^{L} f(x)\overline{g_{n}(x)}dx = <f,g_{n}>$$ The Fourier ..... I have solved all the questions in this question (with the help of many people) except letter (g) I do not know how I will deduce from all the questions before (g) the answer of (g) I do not know exactly which letter before (g) will help me deduce (g), could anyone explain this for me?

Answer 1

g follows from f using the following: \begin{align} \|f\|_{L^2}^2&=\left\|\left(f-\sum_{n=-N}^{N}\hat{f}(n)e^{inx}\right)+\sum_{n=-N}^{N}\hat{f}(n)e^{inx}\right\|_{L^2}^2 \\ &= \left\|f-\sum_{n=-N}^{N}\hat{f}(n)e^{inx}\right\|_{L^2}^2+\sum_{n=-N}^{N}|\hat{f}(n)|^2 \end{align}