Bessel's inequality

Question

Let $f \in L^2(-\pi, \pi)$, $S_{N,f} = \sum_{|k| \leq N, \ k \in \mathbb{Z}}\langle {e_k,f} \rangle e_{k}(x)$ where $e_k(x) = (2\pi)^{-1/2}e^{2ikx}$. Is it true $\|S_{N,f}\|_{L^2} \leq \|f\|_{L^2}$?

I need a hint on where to start on this problem.

Answer 1

Note that $\{e_k\}_{k\in\mathbb{Z}}$ is an orthonormal set in $L^2(-\pi,\pi)$. It follows that $$\|S_{N,f}\|_2^2=\sum_{|k|\leq N}|\langle e_k,f\rangle|^2.$$ Therefore $$\begin{aligned}0&\leq\|f-S_{N,f}\|_2^2=\langle f-S_{N,f},f-S_{N,f}\rangle\\ &=\langle f,f\rangle-\langle S_{N,f},f\rangle-\langle f,S_{N,f}\rangle+\langle S_{N,f},S_{N,f}\rangle\\ &=\langle f,f\rangle-\overline{\langle f,S_{N,f}\rangle}-\langle f,S_{N,f}\rangle+\langle S_{N,f},S_{N,f}\rangle\\ &=\|f\|_2^2+\|S_{N,f}\|_2^2-2\text{Re}\langle f,S_{N,f}\rangle\\ &=\|f\|_2^2+\|S_{N,f}\|_2^2-2\text{Re}\sum_{|k|\leq N}\langle e_k,f\rangle\overline{\langle e_k,f\rangle}\\ &=\|f\|_2^2+\|S_{N,f}\|_2^2-2\text{Re}\sum_{|k|\leq N}|\langle e_k,f\rangle|^2\\ &=\|f\|_2^2+\|S_{N,f}\|_2^2-2\sum_{|k|\leq N}|\langle e_k,f\rangle|^2\\ &=\|f\|_2^2-\|S_{N,f}\|_2^2 \end{aligned}$$