Proposition 7

Tag 0CDI, permanent. Cite it as [Bourbaki, Tag 0CDI].

General Topology, chapter X, § 3. Special function spaces

Let $X, Y, Z$ be three normed spaces over $K$ . The canonical mapping of the normed space $\mathscr{L}(X, Y; Z)$ into the space of linear mappings of $X$ into $\mathscr{L}(Y; Z)$ which sends each $u \in \mathscr{L}(X, Y; Z)$ to the mapping $x \to u(x, .)$ is an isometry of $\mathscr{L}(X; Y; Z)$ onto $\mathscr{L}(X; \mathscr{L}(Y; Z))$ .

This follows immediately from the definitions and the relation

\sup_{\|x\| \leq 1} \left( \sup_{\|y\| \leq 1} \|u(x, y)\| \right) = \sup_{\|x\| \leq 1, \|y\| \leq 1} \|u(x, y)\|.

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

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