must any $\phi \in \operatorname{Hom}_G(V, L^2(G))$ be continuous?

Question

Let $G$ be a compact group and $V$ a finite-dimensional vector space with a continuous $G$-action. Consider a linear map $\phi: V \to L^2(G)$ satisfying that for any $v \in V, h \in G$:

$$ \phi(v)(g h) = \phi(h \cdot v)(g) \quad \text{for almost all $g \in G$} $$ Must every such $\phi$ be continuous?

In my representation theory course we used this fact to conclude that matrix coefficients give rise to an isomorphism $V^* \cong \operatorname{Hom}_G(V, L^2(G))$ and ultimately prove the Peter-Weyl theorem.

EDIT: This wasn't the question I was meaning to ask, so I asked the right question here.

Answer 1

A linear map from a finite-dimensional normed vector space is always continuous. The $G$-equivariance plays no role here.