Fourier series converge in mean square

The characters form an orthonormal basis of $L^2(\mathbb{T};\mathbb{C})$, and for a complete orthonormal family $x$ is the norm limit of the finite-subset net of the partial sums ${\sum }_{i\in F}⟨x,e_i⟩e_i$ (The trigonometric system is complete in $L^2$ of the torus, Fourier expansion in a Hilbert space).

The Fourier coefficient satisfies $\widehat{f}(k)=⟨f,e_k⟩$, so the partial sums displayed above are exactly the values ${\sum }_{k\in F}⟨f,e_k⟩e_k$ of the net at the symmetric index sets $F=[-N,N]$ (Fourier coefficients and trigonometric polynomials on the torus, The trigonometric characters are orthonormal in $L^2$ of the torus).

For every finite $F\subseteq \mathbb{Z}$ there is $N\in \mathbb{N}$ with $F\subseteq \{k:\mathrm{\mid }k\mathrm{\mid }\le N\}$. If $F=\mathrm{\varnothing }$, take $N=0$; otherwise the nonempty finite set $\{\mathrm{\mid }k\mathrm{\mid }:k\in F\}$ has a maximum and one may take that maximum (Every nonempty finite set of reals has a maximum and a minimum).

If a net in a metric space converges to $x$ then every cofinal sub-net converges to $x$: given $\epsilon >0$ the net is eventually in the ball of radius $\epsilon$ around $x$ at some index, and any cofinal sub-net passes beyond that index (Directed preorders and nets, Square-summable families on an arbitrary index set and the space ${\mathrm{\ell }}^2(I)$).