Lemma 2

Let $\pi$ and $\varrho$ be continuous linear representations of G in separated topological K-vector spaces. Suppose that $\pi$ is irreducible. Every nonzero G-morphism from $\pi$ into $\varrho$ is injective, and every nonzero G-morphism from $\varrho$ into $\pi$ has dense image.

In particular, if $\pi$ and $\varrho$ are irreducible and finite-dimensional, then every nonzero G-morphism from $\pi$ into $\varrho$ is an isomorphism.

Since the space of $\varrho$ is separated, the kernel of a G-morphism $u$ from $\pi$ into $\varrho$ is closed, and gives rise to a subrepresentation of $\pi$. If the morphism $u$ is nonzero, its kernel must therefore be reduced to 0, since $\pi$ is an irreducible representation in a separated topological vector space. Similarly, the closure of the image of a nonzero G-morphism from $\varrho$ into $\pi$ is a nonzero subrepresentation of $\pi$, and is therefore equal to the space of $\pi$.

The last assertion follows from the preceding.