Remark

Let E and F be normable topological vector spaces over K. The vector space $\mathscr{L}(E; F)$ of continuous linear mappings from E into F, endowed with the topology of bounded convergence, is a normable topological vector space (TVS, III, p. 14).

Let E and F be normable topological vector spaces over $\mathbf{R}$. The $\mathbf{C}$-linear mapping $\varphi :\mathscr{L}(E; F)_{(\mathbf{C})}\rightarrow \mathscr{L}(E_{(\mathbf{C})}; F_{(\mathbf{C})})$ defined by $\varphi (\lambda \otimes u) =\lambda u_{(\mathbf{C})}$ is an isomorphism of complex topological vector spaces. In particular, the dual of $E_{(\mathbf{C})}$ is identified with the complexified dual of E and the normable algebra $\mathscr{L}(E_{(\mathbf{C})})$ with the complexified of the normable algebra $\mathscr{L}(E)$.

Let S be a compact subset of $\mathbf{C}$ stable under complex conjugation. Consider the $\mathbf{C}$-algebra $\mathscr{O}(S)$ of germs of holomorphic functions with complex values in a neighbourhood of S, endowed with the complex locally convex space structure defined in No. 1 of I, p. 49. If U is an open neighbourhood of S in $\mathbf{C}$, and $h: U\rightarrow \mathbf{C}$ a holomorphic function, the image V of U under complex conjugation is an open neighbourhood of S in $\mathbf{C}$ and $h^*:w\mapsto \overline{h(\overline{w})}$ is a holomorphic function on V. By passing to the inductive limit one deduces a continuous involution $f\mapsto f^*$ in the algebra $\mathscr{O}(S)$. In particular, one has:

for $f, g$ in $\mathscr{O}(S)$ and $\lambda$ in $\mathbf{C}$.

The set of germs $f\in \mathscr{O}(S)$ such that $f=f^*$ is denoted by $\mathscr{O}_{\mathbf{R}}(S)$. It is a closed full sub-$\mathbf{R}$-algebra of $\mathscr{O}(S)$.