Exercise 18, § 4

Suppose that G is compact. Let $\varrho$ and $\varrho '$ be unitary representations of G. The representation $\varrho$ is weakly contained in $\varrho '($cf. exercise 11 of V, p. 497) if and only if for every irreducible unitary representation $\pi$ of G, the condition $m_{\pi}(\varrho )\geqslant 1$ implies that $m_{\pi}(\varrho ')\geqslant 1$. (Reduce to the case where $\varrho '$ is irreducible; for every normalized diagonal matrix coefficient $f$ of $\varrho$, exercise prove, applying exercise 13 of V, p. 498, that there exists a diagonal matrix coefficient of $\varrho '$ which is not orthogonal to $f$, and conclude using the orthogonality relations.)