So let G be a compact group. Peter-Weyl's theorem states that $$L^2(G)= \overline{\bigoplus_{\lambda \in \hat{G}}M_\lambda}. $$ Where $M_\lambda$ is the space generated by the coefficients of a representation in the class $\lambda$. So i am trying to prove that the space $$M = \bigoplus_{\lambda \in \hat{G}}M_\lambda$$ is dense in C(G). For that we use Stone-weiserstrass theorem. So the first step is to prove that $M$ is an algebra. So i am not quite sure what we really have to show here? Just that a product of two elements (matrix coefficients) is contained in $M$, and is that enough? At least thats what i am trying.
So for that we introduce the tensor product of two (finite) representations ($\pi_1, \pi_2)$ on resp. $(\mathcal{H_1}, and\ \mathcal{H_2})$
So i get if $\mathcal{H_1}, \mathcal{H_2}$ are Hilbert spaces, there is an inner product $(\cdot | \cdot)$ $\mathcal{H}_1 \otimes \mathcal{H}_2$. So i get that $$\big( (\pi_1\otimes \pi_2)(g)(u_1 \otimes u_2) | (v_1 \otimes v_2) \big) = ( \pi_1(g)u_1 | v_1)(\pi_2(g)u_2 | v_2).$$ So the product of a coefficient of $\pi_1$ and $\pi_2$ is a coefficient of $\pi_1 \otimes \pi_2$. So how can i conclude from this, that for $\lambda, \mu \in \hat{G}$ the representation $\pi_\lambda \otimes \pi_\mu$ can be decomposed into a sum of irreducible representations $$ \pi_\lambda \otimes \pi_\mu = \bigoplus_{\nu \in E(\lambda, \mu)} c(\lambda, \mu; \nu) \pi_\nu. $$ And why does this implies that $M$ is an algebra??