Corollary

Tag 09CT, permanent. Cite it as [Bourbaki, Tag 09CT].

Integration, chapter VI, Appendix 0. Complements on topological vector spaces

If $\Phi$ is a continuous bilinear form on $E \times F$ , then ${}^l \Phi$ is a continuous linear mapping of $E$ into the strong dual $F'_b$ of $F$ . If, moreover, $E$ and $F$ are normed spaces, then $\|{}^l \Phi\| = \|\Phi\|$ .

The first assertion follows from Prop. 1 and the fact that every neighborhood of 0 in $F'_b$ absorbs every equicontinuous subset of $F'$ . If $E$ and $F$ are normed, then

\|\Phi\| = \sup_{\|x\| \leq 1, \|y\| \leq 1} |\Phi(x, y)| = \sup_{\|x\| \leq 1} \left( \sup_{\|y\| \leq 1} |\langle l \Phi(x), y \rangle| \right) = \sup_{\|x\| \leq 1} \|l \Phi(x)\| = \|l \Phi\|,

whence the second assertion.

Interchanging the roles of E and F, one obtains results analogous to Prop. 1 and its corollary for the linear mappings $r \Phi$ ; we leave to the reader the task of stating them.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Prop. 1