Theorem 2

The Fourier cotransformation of G induces by passing to the subspaces an isometric isomorphism of $L^2(G)$ onto $F^2(\widehat{G})$. Its inverse $\mathscr{F}_G$ associates with an element $(u_{\pi})_{\pi\in\widehat{G}}$ of $F^2(\widehat{G})$ the sum of the series

which converges in $L^2(G)$.

By the Peter-Weyl theorem (th. 1 of V, p. 462), the Hilbert space $L^2(G)$ is the Hilbert sum of the spaces $\boldsymbol{\varrho }_G(\overline{\pi})$ for $\pi \in \widehat{G}$. For every $\pi \in \widehat{G}$, the linear mapping $f\mapsto \pi (f)$ of $\boldsymbol{\varrho }_G(\overline{\pi})$ into End(E$_{\pi}$) is an isometric isomorphism (lemma 4). Consequently, the restriction to $L^2(G)$ of the Fourier cotransformation defines an isometric isomorphism of $L^2(G)$ onto $F^2(\widehat{G})$.

Let $f\in L^2(G)$. Let $\pi \in \widehat{G}$, and let $f_{\overline{\pi}}\in \mathscr{C}(G)$ be the orthogonal projection of $f$ onto $\boldsymbol{\varrho }_G(\overline{\pi})$ (th. 1 of V, p. 462). Since this space is the $\overline{\pi}$-isotypical component of $\boldsymbol{\delta }_G$ (cor. 4 of V, p. 463), one has $f_{\overline{\pi}}=$ dim($\pi$ )$\boldsymbol{\delta }_G(\chi_{\pi})(f)$ by cor. 2 of V, p. 465. For every $x\in G$, this gives

$f_{\overline{\pi}}(x) =$ dim($\pi$ )$\int_G\chi_{\pi}(g)(\boldsymbol{\delta }_G(g)f)(x)d\mu(g)$

= dim($\pi$ )$\int_G\chi_{\pi}(g)f(xg)d\mu(g)$

= dim($\pi$ )$\int_G\chi_{\pi}(x^{-1}y)f(y)d\mu(y)$

= dim($\pi$)Tr($\pi (x^{-1})\pi (f)$) $=\langle \pi (x)|\pi (f)\rangle$,

that is to say $f_{\overline{\pi}}=\mathscr{F}_{\pi}(\pi (f))$.

Since $f$ is the sum in $L^2(G)$ of the family $(f_{\overline{\pi}})$ by the Peter–Weyl theorem, we obtain

where the series converges in $L^2(G)$. This proves the theorem.