Remark 4

For every topological vector space E over the field $\mathbf{R}$, we denote by $E_{(\mathbf{C})}$ the complexified topological vector space of E (EVT, II, p. 65). We identify E with a real topological vector subspace of $E_{(\mathbf{C})}$ by the mapping $x\mapsto 1\otimes x$; the real topological vector space underlying $E_{(\mathbf{C})}$ is then the topological direct sum of E and $iE$.

Let $u$ be a linear mapping from E into a separated topological vector space F over $\mathbf{R}$. Let $u_{(\mathbf{C})}$ denote the $\mathbf{C}$-linear mapping from $E_{(\mathbf{C})}$ into $F_{(\mathbf{C})}$ which extends $u$. The sets of the form $V +iV$, where V is a neighbourhood of 0 in E, form a fundamental system of neighbourhoods of 0 in $E_{(\mathbf{C})}$. In order that the set $u_{(\mathbf{C})}(V +iV) =u(V) +iu(V)$ be relatively compact in $F_{(\mathbf{C})}$, it is necessary and sufficient that $u(V)$ be relatively compact in F. Consequently, in order that the $\mathbf{C}$-linear mapping $u_{(\mathbf{C})}$ be compact, it is necessary and sufficient that the $\mathbf{R}$-linear mapping $u$ be so.

In the same order of ideas, when $v$ is a continuous $\mathbf{C}$-linear mapping from $E_{(\mathbf{C})}$ into a $\mathbf{C}$-topological vector space G, in order that $v$ be compact it is necessary and sufficient that its restriction to E be compact.