Peter–Weyl gives density, not finite equality

By [L2] every finite-dimensional continuous representation of $S^1$ is a direct sum of characters $z\mapsto z^n$, so all of its matrix coefficients are finite linear combinations of those characters. Hence every finite sum of matrix coefficients is a function of the form $\theta \mapsto p(e^{i\theta })$ for a Laurent polynomial $p$, which is smooth in the real variable $\theta$.

The continuous function $f(e^{i\theta }):=\mathrm{\mid }\theta \mathrm{\mid }$ for $\theta \in (-\pi ,\pi ]$ on $S^1$ is not differentiable at $\theta =0$, while every function $\theta \mapsto p(e^{i\theta })$ with $p$ a Laurent polynomial is differentiable there; hence $f$ is not a finite sum of matrix coefficients.

On the other hand, by [L1] the finite sums of matrix coefficients are uniformly dense, so $f$ is a uniform limit of such sums; Peter–Weyl therefore gives density, not finite equality, and the statement of this item is false.