Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
If $G$ is an arbitrary monoid with multiplication $m:G\times G\to G$ then $m$ induces a map $m^*:Fun(G,k)\to Fun(G\times G,k)$, $m^*(f):f\mapsto f\circ m$. We say that $f\in Fun(G,k)$ is representative if $m^*(f)$ is in the image of the canonical map $Fun(G,k)\otimes Fun(G,k)\hookrightarrow Fun(G\times G,k)$. Equivalently, $f$ is representative if the span of all functions $g\cdot f : h\mapsto f(h\cdot g)$ is finite dimensional. It follows then that $m^*(f)$ is in fact in (the image of) $R(G)\otimes R(G)$ where $R(G)$ is the space of all representative functions on $G$.
Peter-Weyl theorem says that the continuous representative functions form a dense subspace of the space of all continuous functions on a compact Lie group $G$.
1 to 1 of 1