Proposition 1

— The mapping

of $\widetilde{\mathscr{K}}(X; E) \times \mathscr{M}(X; \mathbf{C})$ into $E'*$ is bilinear.

The proposition follows immediately from Def. 1 of No. 1.

Let $u$ be a continuous linear mapping of $E$ into a locally convex space $F$; we know that the transpose $^t u$ is a linear mapping of the dual $F'$ of $F$ into the dual $E'$ of $E$; we shall denote by $^{tt} u$ the mapping ${E'}^* \to {F'}^*$, the transpose of $^t u$ (in the algebraic sense); when $E$ and $F$ are Hausdorff and are canonically identified with subspaces of ${E'}^*$ and ${F'}^*$, respectively, $^{tt} u$ extends the mapping $u$. With these notations: