Riesz–Fischer: the Fourier coefficient map is onto the space of square-summable families

By [A1] the characters are an orthonormal basis of $L^2(\mathbb{T};\mathbb{C})$, and by [A3] the map $\mathrm{\Phi }$ is exactly the coefficient map of that basis.

The general coefficient-isometry theorem for Hilbert spaces with a given orthonormal basis therefore applies to $\mathrm{\Phi }$: it is a linear bijection onto ${\mathrm{\ell }}^2(\mathbb{Z},\mathbb{C})$ satisfying $\mathrm{\parallel }\mathrm{\Phi }(f){\mathrm{\parallel }}_2=\mathrm{\parallel }f{\mathrm{\parallel }}_2$ and $⟨\mathrm{\Phi }(f),\mathrm{\Phi }(g){⟩}_{{\mathrm{\ell }}^2}=⟨f,g⟩$, and the target space is complete.

In particular $\mathrm{\Phi }$ is surjective: for every $a\in {\mathrm{\ell }}^2(\mathbb{Z},\mathbb{C})$ there is exactly one $f\in L^2(\mathbb{T};\mathbb{C})$ with $\widehat{f}(k)=a_k$ for all $k$. For this $f$, [A5] says that the finite-subset net ${\sum }_{k\in F}\widehat{f}(k)e_k$ converges to $f$. Given a finite $F_0\subseteq \mathbb{Z}$, some $N_0$ has $F_0\subseteq \{-N_0,\dots ,N_0\}$; hence the symmetric finite sets are cofinal, and the symmetric sums ${\sum }_{\mathrm{\mid }k\mathrm{\mid }\le N}{a}_k{e}_k$ converge to $f$.

Steps 2.1 and 3.1 are the announced surjective isometry and the Riesz–Fischer uniqueness of the class realizing a given square-summable coefficient family.