Remark

Tag 07MQ, permanent. Cite it as [Bourbaki, Tag 07MQ].

— 3) Let A be a dense subset of the unit ball of E. In view of the continuity of $u$ , we also have

r(u) = \sup_{x \in A} q(u(x)) .

For example

r(u) = \sup_{p(x) < 1} q(u(x)) .

Since we have $u(tx) = tu(x)$ for $t \in \mathbf{R}$ , we also have,

r(u) = \sup_{p(x) = 1} q(u(x)) = \sup_{p(x) \neq 0} \frac{q(u(x))}{p(x)} .

whenever $p \neq 0$ .

4) The formula (2) shows that the map $u \mapsto r(u)$ is lower semi-continuous on $\mathscr{L}_s(E; F)$ .

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

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