Representative functions on a compact group

Assume the Axiom of Choice (The Axiom of Choice). Let $K$ be a compact Hausdorff topological group (Topological group: multiplication and inversion are continuous, Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right, Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not) with normalized Haar probability measure $\mu$ (Normalized Haar probability on a compact group). A function $f:K\to \mathbb{C}$ is a representative function when there are finitely many finite-dimensional continuous unitary representations ${\pi }_1,\dots ,{\pi }_r$ of $K$ (Strongly continuous unitary representations, invariant linear subspaces and intertwiners), vectors $v_j,w_j$ in the carrier of ${\pi }_j$ and scalars $c_j\in \mathbb{C}$ with $f={\sum }_{j=1}^r{c}_j{c}_{v_j,w_j}^{{\pi }_j},c_{v,w}^{\pi }(k)=⟨\pi (k)v,w⟩,$ the matrix coefficient convention of Matrix coefficient of a unitary representation, which is linear in $v$ and conjugate-linear in $w$. We write $R(K)\subseteq C(K,\mathbb{C})$ for the set of representative functions; by definition it is the linear span (Linear subspace of a vector space) of the matrix coefficients of the continuous finite-dimensional unitary representations of $K$, and each such coefficient is a continuous function by Matrix coefficient of a unitary representation, so the inclusion in $C(K,\mathbb{C})$ is well defined.

Nonunitary finite-dimensional representations give nothing new. Let $\rho$ on $V$ be any continuous finite-dimensional complex representation of $K$, unitarizable by Averaging a Hermitian form unitarizes a finite-dimensional compact-group representation: there is an inner product $h$ on $V$ making $\rho$ unitary. Fix any inner product $h_0$ on $V$ with orthonormal basis $e_1,\dots ,e_d$ and expand $\rho (k)e_j={\sum }_i{\rho }_{ij}(k)e_i$. The functions ${\rho }_{ij}(k)=h_0(\rho (k)e_j,e_i)$ are $h_0$-matrix coefficients of $\rho$, and expanding $v={\sum }_j{a}_j{e}_j$ gives $h(\rho (k)v,w)={\sum }_{i,j}{a}_{j}h(e_i,w){\rho }_{ij}(k),$ a finite linear combination of the ${\rho }_{ij}$ with scalars independent of $k$; the converse containment is the same computation run with the roles of $h$ and $h_0$ exchanged. Hence $R(K)$ is also the linear span of the matrix coefficients of all continuous finite-dimensional complex representations of $K$. No closure, completeness, density or point-separation property is asserted here.