$f \in L^2[-\pi,\pi]$ implies squares of Fourier coefficients are summable?

Question

Suppose $f \in L^2[-\pi,\pi]$. I want to show that $\sum_{-\infty}^{\infty} a_n^2 < \infty$ where $a_n$ is the $n$-th Fourier coefficient of $f$. I saw Rudin's real and compleax book. I was unable to make out the proof. Any concrete solution will be appreciated.

Answer 1

This is Bessel's inequality. For an integer $N$ let $$g_N(x)=\sum_{n=-N}^N a_n e^{in x}$$ and $$h_N(x)=f(x)-g_N(x).$$ Prove that $g_N$ and $h_N$ are orthogonal: $\int_{-\pi}^\pi\overline{g_N(x)}h_N(x)\,dx=0$. Then prove that $$\int_{-\pi}^\pi|f(x)|^2\,dx =\int_{-\pi}^\pi|g_N(x)|^2\,dx+\int_{-\pi}^\pi|h_N(x)|^2\,dx \ge \int_{-\pi}^\pi|g_N(x)|^2\,dx$$ and this is $2\pi\sum_{n=-N}^N|a_n|^2$.