Proposition 9

Let G be a topological group and let $\varrho$ be a unitary representation of G in a complex Hilbert space E. Let $\pi$ be an irreducible unitary representation of G in a complex Hilbert space $E_{\pi}$. Let us denote by $v$ the linear mapping of Hom$_G(\pi , \varrho )\otimes E_{\pi}$ into E such that $v(u\otimes x) =u(x)$ for every $(u, x)\in$ Hom$_G(\pi , \varrho )\times E_{\pi}$.

The linear mapping $v$ is injective, and its image is the sum of the spaces of all the subrepresentations of $\varrho$ which are isomorphic to $\pi$. In particular, the image of $v$ is dense in $M_{\pi}(\varrho )$.

Cor. 5 of V, p. 388 implies that the image of $v$ is the sum of the spaces of all the subrepresentations of $\varrho$ which are isomorphic to $\pi$.

Let us prove that $v$ is injective. Let $(u_i)_{i\in I}$ be a basis of the vector space Hom$_G(\pi , \varrho )$. For $i\in I$, let us denote by $F_i$ the image of $u_i$ and by $\widetilde{u}_i: E_{\pi}\rightarrow F_i$ the G-isomorphism deduced from $u_i$ by passing to the subspaces.

Let us first prove, by induction on the cardinal of a finite subset J of I, that the subspaces $(F_i)_{i\in J}$ are in direct sum.

The case where J is empty is immediate. Suppose that J is nonempty and that the required property holds for the subsets of I of cardinal at most Card(J) $-1$.

Let $j$ be a fixed element of J. The induction hypothesis implies that the subspaces $F_i$ for $i\in J-\{j\}$ are in direct sum. Let $F'$ be their sum; it remains now to prove that $F_j\cap F'=\{0\}$.

Suppose that the intersection of $F_j$ and $F'$ is not reduced to 0; this intersection is a subrepresentation of $F_j$, and since the latter is irreducible, we therefore have $F_j\cap F'= F_j$, whence $F_j\subset F'$.

For $i\in J-\{j\}$, denote by pr$_i: F'\rightarrow F_i$ the projection and by $\iota_i: F_i\rightarrow F'$ the inclusion. We have a canonical isomorphism

Hom$_G(F_j,F')\rightarrow \bigoplus_{i\in J-\{j\}}$ Hom$_G(F_j,F_i)$

(formula (1), p. 377) such that $u: F_j\rightarrow F'$ has as image the family (pr$_i\circ u$)$_{i\in J-\{j\}}$ (A, II, p. 13, cor. 1). Corollary 2 of V, p. 387 implies that $\widetilde{u}_i\circ \widetilde{u}^{-1}_j$, which is nonzero, is a basis of the space Hom$_G(F_j,F_i)$. Consequently, the family of G-morphisms $(\iota_i\circ \widetilde{u}_i\circ \widetilde{u}^{-1}_j)_{i\in J-\{j\}}$ is a basis of Hom$_G(F_j,F')$.

Denote by $\iota$ the inclusion of $F_j$ in $F'$; it is an element of Hom$_G(F_j,F')$, hence it is a linear combination of the G-morphisms $\iota_i\circ \widetilde{u}_i\circ \widetilde{u}^{-1}_j$ for $i\in J-\{j\}$. It follows that $u_j$ is a linear combination of the mappings $u_i$ for $i\not =j$, which contradicts the linear independence of the family $(u_i)_{i\in I}$. The assertion is therefore proved by induction.

Let us now prove that $v$ is injective. Let $w$ be an element of Hom$_G(\pi , \varrho )\otimes E_{\pi}$. There exists a unique family $(x_i)_{i\in I}$ in $E_{\pi}$ with finite support such that

(A, II, p. 62, cor. 1) and we then have

From what precedes, the condition $v(w) = 0$ therefore implies that $u_i(x_i) = 0$ for every $i\in I$, whence $x_i= 0$ for every $i$ since $u_i$ is injective, and therefore $w= 0$.