Remark

If $K =\mathbf{R}$ and if $\varrho$ is a continuous linear representation of G in an $\mathbf{R}$-vector space E, then the mapping $\varrho_{(\mathbf{C})}:g\mapsto 1\otimes \varrho (g)$ is a continuous linear representation of G in $E_{(\mathbf{C})}$; if $\varrho$ is bounded, then $\varrho_{(\mathbf{C})}$ is bounded.

For every topological K-vector space E, the homomorphism which associates $1_E$ with every $g\in G$ is a continuous linear representation of G in E, called the trivial representation of G in E.

Let H be a subgroup of G. The restriction of $\varrho$ to H is a continuous linear representation of H in E, denoted Res$^G_H(\varrho )$.

Let $\varrho_1$ and $\varrho_2$ be continuous linear representations of G in topological K-vector spaces $E_1$ and $E_2$. A G-morphism $u$ of $\varrho_1$ into $\varrho_2$ is a morphism of linear representations which is continuous, that is to say a continuous linear mapping $u: E_1\rightarrow E_2$ such that $u\circ \varrho_1(g) =\varrho_2(g)\circ u$ for every $g\in G$. The vector space of G-morphisms of $\varrho_1$ into $\varrho_2$ is denoted by Hom$_G(\varrho_1, \varrho_2)$. It is a closed subspace of the space $\mathscr{L}(E_1; E_2)$ endowed with the topology of simple convergence.

Let $\varrho$ be a continuous linear representation of a group G in a topological K-vector space E. The identity mapping $1_E$ of E is a G-morphism of $\varrho$ into $\varrho$, denoted by $1_{\varrho}$. If $\varrho_1,\varrho_2$ and $\varrho_3$ are continuous linear representations of G in topological K-vector spaces $E_1, E_2, E_3$ respectively, and if $u: E_1\rightarrow E_2$ and $v: E_2\rightarrow E_3$ are G-morphisms, then $v\circ u$ is a G-morphism.

A G-morphism $u$ from $\varrho_1$ into $\varrho_2$ is a G-isomorphism if there exists a G-morphism $v$ from $\varrho_2$ into $\varrho_1$ such that $v\circ u= 1_{\varrho_1}$ and $u\circ v= 1_{\varrho_2}$. It is necessary and sufficient for this that $u$ be a G-morphism and an isomorphism of topological vector spaces from the space of $\varrho_1$ onto the space of $\varrho_2$; its inverse $v=u^{-1}$ is then in fact a G-morphism. If there exists a G-isomorphism from $\varrho_1$ onto $\varrho_2$, one says that these representations are isomorphic.