Proposition 18

The complex topological vector space $E_{(\mathbf{C})}$ is normable (resp. complete) if and only if E is normable (resp. complete).

The underlying real topological vector space of $E_{(\mathbf{C})}$ is isomorphic to $E\times E$. Hence $E_{(\mathbf{C})}$ is complete if and only if E is complete, and E is normable if $E_{(\mathbf{C})}$ is so.

Suppose conversely that E is normable. Let $p$ be a norm defining the topology of E and B the unit ball of $p$. There exists a closed balanced neighbourhood V of 0 in $E_{(\mathbf{C})}$ contained in $B +iB$ (TVS, II, p. 66). The sets $\lambda V$, where $\lambda$ ranges over $\mathbf{R}^*_+$, therefore form a fundamental system of neighbourhoods of 0 in $E_{(\mathbf{C})}$. The gauge of V is a norm on $E_{(\mathbf{C})}$ defining the topology of $E_{(\mathbf{C})}$, therefore $E_{(\mathbf{C})}$ is normable.