Definition 10

Let G be a topological group. Let $\varrho$ be a unitary representation of G in a complex Hilbert space E and $\pi$ an irreducible unitary representation of G in a complex Hilbert space.

The multiplicity of $\pi$ in $\varrho$ is defined to be the cardinal of the set I for any family $(E_i)_{i\in I}$ of closed subspaces of E stable under G such that the subrepresentation of $\varrho$ induced in $E_i$ is isomorphic to $\pi$ for every I and such that $M_{\pi}(\varrho )$ is the Hilbert sum of the subspaces $E_i$.

If the multiplicity of $\pi$ in $\varrho$ is finite, then it is equal to the dimension of the space Hom$_G(\pi , \varrho )$ (resp. to the dimension of Hom$_G(\varrho , \pi )$) by formula (1) of V, p. 377 and corollary 2 of V, p. 387. This is not always the case in general.