Let $f: [0,2\pi] \to \mathbb C$ be a riemann integrable function.
Let $F_{N}: \mathbb C^{2N+1} \to \mathbb R$,
$F_{N}(\gamma_{-N},...,\gamma_{N}):=\vert\vert f - \sum^{N}_{k=-N}\gamma_{k}\xi_{k}\vert\vert_{2} $, whereby $\xi_{k}: [0,2\pi]\to \mathbb C, \xi_{k}:=e^{ikx}/\sqrt{2\pi}$.
I am required to prove that the minimum value of $F_{N}$ only gets "taken on"/defined in:
($\sqrt{2\pi}c_{-N},...,\sqrt{2\pi}c_{N})$
where $(c_{k})_{k\in\mathbb Z}$ are the fourier coefficients of $f$.
Ideas:
$\vert\vert \sum^{\infty}_{k=N+1}c_{k}e^{ikx}+\sum^{-N-1}_{k=-\infty}c_{k}e^{ikx}+\sum^{N}_{k=-N}c_{k}e^{ikx}+\sum^{N}_{k=-N}\gamma_{k}\frac{e^{ikx}}{\sqrt{2\pi}}\vert\vert_{2}=\vert\vert \sum^{\infty}_{k=N+1}c_{k}e^{ikx}+\sum^{-N-1}_{k=-\infty}c_{k}e^{ikx}+\sum^{N}_{k=-N}e^{ikx}(c_{k}-\frac{\gamma{k}}{\sqrt{2\pi}})\vert\vert_{2}$
and $\sum^{N}_{k=-N}e^{ikx}(c_{k}-\frac{\gamma{k}}{\sqrt{2\pi}})=0 \ \Longleftrightarrow \gamma_{k}=\sqrt{2\pi}c_{k}$ $\forall k \in \{-N,...,N\}$ and therefore $F_{N}(\gamma_{-N},...,\gamma_{N}):=\vert\vert f - \sum^{N}_{k=-N}\gamma_{k}\xi_{k}\vert\vert_{2}$ is minimized.
Is this a sufficient proof seeing as though I have made no reference to the norm itself?
Write $f$ in terms of the Fourier coefficients, then write explicitly what the norm is. You will notice that it will be a sum of squares with terms of type $(c_k-\gamma_k)^2$. The sum of positive terms is minimum when all those terms are $0$.