Definition 1

Let $\varrho$ be a continuous linear representation of a topological group G in a topological vector space E. A continuous linear representation $\pi$ of G in F is a subrepresentation of $\varrho$ if F is a closed subspace of E and if, for every $g\in G$, the space F is stable under $\varrho (g)$ and $\pi (g)$ is the endomorphism of F deduced from $\varrho (g)$ by passing to subspaces.

Let $\varrho$ be a continuous linear representation of a topological group G in a topological vector space E. The closure of $\{0\}$ in E is a subrepresentation of $\varrho$. More generally, the closure of a subspace stable under the endomorphisms $\varrho (g)$ is a subrepresentation of $\varrho$.

A subrepresentation $\pi$ of $\varrho$ is determined in a unique manner by a closed subspace F, stable under all the endomorphisms $\varrho (g)$. This latter condition will often be stated in the form “F is a G-invariant subspace of E”. One will then also say that F defines a subrepresentation of $\varrho$, or sometimes, by abuse of language, that F is a subrepresentation of $\varrho$, or of E.

Let F be a subspace of E defining a subrepresentation $\pi$ of $\varrho$. For every $g\in G$, the linear mapping $\varrho (g)$ defines, by passing to the quotient, an endomorphism $\widetilde{\varrho}(g)$ of $E/F$. The mapping $g\mapsto \widetilde{\varrho}(g)$ is a continuous linear representation of G in $E/F$ and the canonical projection of E onto $E/F$ is a G-morphism. One says that $\widetilde{\varrho}$ is the quotient representation of G on $E/F$; it is also denoted by $\varrho /\pi$.

The subspace of the elements of E invariant under the action of G is a trivial subrepresentation of $\varrho$. It is denoted by $\varrho^G$ or else $E^G$.

Let A be a subset of E. The closed subspace F generated by the elements $\varrho (g)x$, where $g\in G$ and $x\in A$, is a subrepresentation of $\varrho$, called the subrepresentation of $\varrho$ generated by A; if F = E, one says that A generates $\varrho$. If the subrepresentation F is finite-dimensional, one says that the subset A is G-finite.

Suppose that A is reduced to a single element $x\in E$. If A generates $\varrho$, one says that $x$ is a cyclic vector of $\varrho$ and that $\varrho$ is a cyclic representation; one says that $x$ is a G-finite vector if A is G-finite.

The set of G-finite vectors of $\varrho$ is a vector subspace of E stable under $\varrho$; it is not necessarily closed.

Let $(\varrho_i)_{i\in I}$ be a family of continuous linear representations of G in locally convex topological K-vector spaces $(E_i)_{i\in I}$. Let E be the direct sum space of the spaces $E_i$. The mapping $g\mapsto (\varrho_i(g))$ is a linear representation of G in E, called the sum, or direct sum, of the representations $\varrho_i$. If I is finite, it is continuous.

If the representations $\varrho_i$ are all equal to a representation $\varrho$ for every $i\in I$, one says that the direct sum of the representations $\varrho_i$ is the sum of Card(I) copies of the representation $\varrho$, and it is also denoted by $\varrho^{Card(I)}$, or Card(I) $\varrho$.

Let $(\varrho_i)_{i\in I}$ and $(\pi_j)_{j\in J}$ be finite families of continuous linear representations of G in topological K-vector spaces $(E_i)_{i\in I}$ and $(F_j)_{j\in J}$, respectively. The canonical isomorphism of K-vector spaces

Hom$_K(\bigoplus_{i\in I}E_i,\bigoplus_{j\in J}F_j)\rightarrow \bigoplus_{(i,j)\in I\times J}$ Hom$_K(E_i,F_j)$ (A, II, p. 13, Corollary 1) induces, by passing to the subspaces, an isomorphism

(1) Hom$_G(\bigoplus_{i\in I}\varrho_i,\bigoplus_{j\in J}\pi_j)\rightarrow \bigoplus_{(i,j)\in I\times J}$ Hom$_G(\varrho_i, \pi_j)$

which is likewise said to be canonical.

Let $\varrho_1$ and $\varrho_2$ be linear representations of G in topological K-vector spaces $E_1$ and $E_2$. For $u\in \mathscr{L}(E_1; E_2)$ and $g\in G$, one sets $\varrho (g)u=\varrho_2(g)\circ u\circ \varrho_1(g^{-1})$. The mapping $g\mapsto \varrho (g)$ is a linear representation of G in $\mathscr{L}(E_1; E_2)$. The space of the elements invariant under this representation coincides with Hom$_G(\varrho_1, \varrho_2)$.

Let $\varrho$ be a continuous linear representation of G in a locally convex space E. Recall (INT, VIII, p. 131, § 2, n$^o2$) that the contragredient representation $\breve{\varrho}$ of $\varrho$ is the linear representation of G in the dual $E'$ of E defined by $\breve{\varrho}(g) =^t\varrho (g^{-1})$.