Fourier series on a torus as Peter–Weyl

Let $\pi :T^r\to U(V)$ be a finite-dimensional continuous irreducible representation. Since $T^r$ is abelian, every $\pi (t)$ commutes with every $\pi (s)$ and hence belongs to ${\mathrm{End}}_{T^r}(V)$; by [L4], every $\pi (t)$ is scalar. Thus every linear subspace of $V$ is invariant, so irreducibility and $V\mathrm{\ne }0$ force $\dim V=1$. Therefore $\pi$ is a character, and [L3] computes the characters: the quotient exponential ${\mathbb{R}}^r\to {\mathbb{R}}^r\mathrm{/}{\mathbb{Z}}^r$ has kernel ${\mathbb{Z}}^r$, so its allowed differentials are exactly $\lambda (X)=2\pi i{\sum }_j{n}_j{X}_j$ with $n_j\in \mathbb{Z}$. Thus the characters are exactly the $e_n$; distinct integer vectors have distinct differentials. Conversely each $e_n$ is a continuous unitary one-dimensional representation and hence irreducible.

Peter–Weyl [L2] therefore says exactly that the one-dimensional representations $e_n$, with their sole normalized matrix coefficient exactly $e_n$, form an orthonormal Hilbert basis of $L^2(T^r)$ and that the regular representation is their Hilbert direct sum weighted by dimension one. In the fixed left-action convention $L_y{e}_n=e_n(-y)e_n$, so the coefficient line has type $e_{-n}$; negation permutes ${\mathbb{Z}}^r$ and every character still occurs once.

For $r\ge 1$, [L1] gives the Fourier expansion and Parseval identity for precisely the basis identified in step 2.1. For $r=0$, the torus is a singleton, its normalized Haar measure has mass one at that point, and ${\mathbb{Z}}^0$ consists of the empty tuple alone. Its sole character is $e_0=1$ and $L^2(T^0)=\mathbb{C}$ with basis $1$; the expansion is $f=f(e)1$ and Parseval is $\mathrm{\parallel }f{\mathrm{\parallel }}_2^2=\mathrm{\mid }f(e){\mathrm{\mid }}^2$. Thus the zero-rank case is proved directly without applying [L1] outside its scope. The AC assumptions of the suppliers are covered by [A1].