Remark 2

Let $E$ be a locally convex space over $\mathbf{C}$, and let $E_0$ be the locally convex space over $\mathbf{R}$ underlying $E$; we know that the mapping $z' \mapsto \Re z'$ which, to every continuous (complex) linear form $z'$ on $E$, makes correspond the continuous (real) linear form $z \mapsto \Re \langle z, z' \rangle$ on $E_0$, is an $\mathbf{R}$-isomorphism of the dual $E'$ onto the dual $E'_0$ of $E_0$ (TVS, II, §8, No. 1). Similarly, the algebraic dual $E'_0*$ of the real vector space $E'_0$ may be canonically identified with the real space underlying the algebraic dual $E'*$ of $E'$. It follows that if $\mu$ is a real measure and $f$ a mapping in $\widetilde{\mathscr{K}}(X; E)$, the formula (1) is again valid when $f$ is regarded as taking its values in $E_0$ and the canonical bilinear forms figuring in the two members as being, respectively, relative to the duality between $E'_0$ and $E'_0*$ for the first member and the duality between $E_0$ and $E'_0$ for the second.