I'm trying to understand the Peter Weyl theorem which provides an orthonormal basis for the space $L^2(G)$ where $G$ is a compact group.
Let $\rho$ be a unitary representation of a compact group, this means that $\rho(g)^*\rho(g)=\rho(g)\rho(g)^* = I \, \, \forall g \in G$.
Denote the matrix elements of $\rho$ as $\phi_{e_i,e_j} = \langle \rho(g)e_i, e_j \rangle$.
The first part of the theorem relates the set of matrix elements and spaces of functions on $G$.
In particular, I'm trying to understand what does it mean that the linear span of the set of matrix elements of unirreps of $G$ is dense in the space of continuous complex-valued functions on $G$, under uniform norm.
So does this mean that
$$\overline{\left\{\sum_{i,j} \langle \rho(g)e_i, e_j \rangle \right\}}^{\|\cdot\|_{\infty}} = L^2(G) \, \, \, ?$$
Also in the above equation the representation $\rho$ seems to be fixed, while I may have to consider all the unirreps of $G$. I have been reading these things from the following document (page 12) and pretty confused about this.