Lemma 9

Let E be a Hilbert space over K. Let $F\subset E$ be a vector subspace dense in E. Then the Hilbert dimension of E is less than or equal to the dimension of F.

If E is of finite Hilbert dimension, this is equal to the dimension of E. Suppose that E is of infinite Hilbert dimension. Then the subspace F is of infinite dimension. Let B be an orthonormal basis of E and let $B'$ be a basis of F. For every $x\in B$, there exists an element $f(x)\in B'$ such that $\langle x|f(x)\rangle \not = 0$, since otherwise one would have $x\in F^{\circ}=\{0\}$. Thus one defines a mapping $f: B\rightarrow B'$.

For every $y\in B'$, the set $\overset{-1}{f}(y)$ is contained in the set of $x\in B$ such that $\langle x|y\rangle \not = 0$, hence is denumerable (EVT, V, p. 21, Prop. 4). By E, III, p. 50, Prop. 4, one obtains Card(B) = Card($f(B)$)$\leqslant$ Card(B$'$).

Let us denote by $Is_G(\pi_1, \pi_2)$ the relation

“G is a topological group and $\pi_1,\pi_2$ are isomorphic unitary representations

of G in Hilbert spaces over K”.

With respect to $\pi_1$ and $\pi_2$, this is an equivalence relation. For every unitary representation $\pi$ of G in a Hilbert space over K, we denote by cl($\pi$ ) the equivalence class of $\pi ($cf. E, II, p. 47); thus it is a unitary representation of G isomorphic to $\pi$; one says that cl($\pi$ ) is the class of $\pi$. Unitary representations $\pi_1$ and $\pi_2$ in Hilbert spaces over K are isomorphic if and only if cl($\pi_1$) $=$ cl($\pi_2$).

Let G be a topological group. Let $\mathfrak{c}$ be a cardinal. The relation

" $\lambda$ is a class of unitary representation of G in

a complex Hilbert space of Hilbert dimension $\leqslant \mathfrak{c}$ "

is collectivizing in $\lambda$ (E, II, p. 3). In fact, every Hilbert space over K of dimension $\leqslant \mathfrak{c}$ is isometrically isomorphic to a Hilbert subspace of $\ell^2(\mathfrak{c})$ (EVT, V, p. 23, cor. 2), and the assertion then follows from E, II, p. 47.

Let $\pi$ be an irreducible unitary representation of G in a Hilbert space E. Let $x$ be a nonzero element of E. Since $\pi$ is irreducible, the vector $x$ is a cyclic vector of $\pi$, which implies that the Hilbert dimension of E is $\leqslant$ Card(G) (lemma 9, applied to the dense subspace generated by the elements $\pi (g)x$ for $g\in G$). The classes of irreducible unitary representations of G in a Hilbert space over K therefore belong to the set of classes of unitary representations of G in a Hilbert space over K of Hilbert dimension $\leqslant$ Card(G) ; consequently, they form a set.