$$$$

For these problems $(X,\mathcal{B},\mu )$ is the measure space where $X=[0,2\pi ]$ and $\mathcal{B}$ is the Borel σ-algebra on $[0,2\pi ]$ and $\mu$ is the restriction of Lebesgue measure to $[0,2\pi ]$.

Take for granted that if $f:[0,2\pi ]\to \mathbb{R}$ is continuous then its integral with respect to $\mu$ equals its Riemann integral on $[0,2\pi ]$.

1. Verify that ${\varphi }_1(x)=1$ and ${\varphi }_{2n+1}(x)=\cos (nx)$ and ${\varphi }_{2n}(x)=\sin (nx)$ both belong to ${\mathsf{L}}^{\mathsf{2}}([0,2\pi ],\mathcal{B},\mu )$ for all $n\in \mathbb{N}$.
2. Check that $$⟨f,g⟩={\displaystyle \frac{1}{2\pi }}\int f\cdot g\mathsf{d}\mu$$ defines an inner product on ${\mathsf{L}}^{\mathsf{2}}([0,2\pi ],\mathcal{B},\mu )$.
3. Check that ${\varphi }_0,{\varphi }_1,{\varphi }_2,\dots$ is an orthonormal collection of functions for the above inner product.

Define $$(\mathcal{F}f)(n)=⟨f,{\varphi }_n⟩$$ for all $n\ge 0$ and all $f$ in ${\mathsf{L}}^{\mathsf{2}}([0,2\pi ],\mathcal{B},\mu )$.

4. By expanding $${\Vert f-\sum _{n=0}^N⟨f,{\varphi }_n⟩{\varphi }_n\Vert }_{\mathsf{2}}^{\mathsf{2}}$$ prove Bessel's inequality that $$\sum _{n=0}^N|(\mathcal{F}f)(n){|}^2\le \Vert f{\Vert }_{\mathsf{2}}^{\mathsf{2}}$$ for all $N\in \mathbb{N}$ and conclude $\mathcal{F}f$ always belongs to ${\ell }^{\mathsf{2}}(\mathbb{N}\cup \{0\})$.