Let $ \left \{X_n\right \} ^{\infty}_{n=1}$ be any orthogonal (in the $L^2$ sense) set of functions. Let $$S_N(f) = \sum^{N}_{n=1} \frac{(f, X_n)}{ \left \|X_n\right \|^2} X_n$$
be the “Fourier series” with respect the $X_n$’s. Show that $S_N(f)$ is linear in $f$ and that $$\left \| S_N(f)\right \| ≤ \left \| f \right \|$$ In other words, taking the Fourier series of a function does not make the norm bigger.
I have no idea how to start this problem. I don't know what it means to "Show that $S_N(f)$ is linear in $f$ "
Linear means that $S_{N}(\alpha f+\beta g)=\alpha S_{N}(f)+\beta S_{N}(g)$, which holds in your case because of the linearity of the inner product in the first coordinate. For the second part, write $$ f = \{f-S_N(f)\}+S_{N}(f) $$ Verify that $(f-S_{N}(f),X_n)=0$ for $n=1,2,\cdots N$. Therefore $(f-S_N(f),S_N(f))=0$ which, by the Pythagorean Theorem, gives $$ \|f\|^2 = \|f-S_N(f)\|^2+\|S_N(f)\|^2 \ge \|S_N(f)\|^2. $$