Definition 9

Let G be a topological group and let $\pi$ be an irreducible continuous representation of G. Let $\varrho$ be a continuous representation of G in a separated locally convex space E. The $\pi$-isotypical component of $\varrho$ is the closure in E of the sum of the spaces of all the subrepresentations of $\varrho$ isomorphic to $\pi$. This subspace is denoted by $M_{\pi}(\varrho )$.

The space $M_{\pi}(\varrho )$ is a closed subspace of E, which defines a subrepresentation of $\varrho$. This space depends only on the class of $\pi$ in $\widehat{G}$.