Theorem Let $G$ and $H$ be compact groups. Let $ρ$ be a finite dimensional irreducible continuous representation of $G×H$ over the field of complex numbers. Then $ρ$ is a tensor product of irreducible representations of $G$ and $H$.
In his book L'integration dans les groupes topologiques, Weil proved this theorem under more general conditions. His proof was short and elementary. He used no functional analysis. On the other hand, Pontryagin proved the same theorem using the Peter-Weyl theorem in his famous book.
I was puzzled. Is Weil's proof correct?