Proposition 7

Let E be a Banach space and F a normed space. The mapping $u\mapsto ((u))$ from $\mathscr{L}(E; F)$ into $\overline{\mathbf{R}}$ is upper semi-continuous.

Let $u\in \mathscr{L}(E; F)$. We have to prove that for every real number $c >((u))$, the set of elements $v\in \mathscr{L}(E; F)$ such that $((v))< c$ is a neighbourhood of $u$. Let B denote the set of elements of E of norm $<1$. By Prop. 6, there exists $y\in E$ such that $u(y)\notin u(B)$ and $\|u(y)\|< c$. The distance $d$ from $u(y)$ to the closed set $u(B)$ is strictly positive. The set V of elements $v$ of $\mathscr{L}(E; F)$ satisfying the relations $\|v(y)\|< c$ and $\|u-v\|(1 +\|y\|)< d$ is a neighbourhood of $u$ in $\mathscr{L}(E; F)$. Let $v\in V$. For every $x\in B$, one has

Consequently $v(y)$ does not belong to $v$(B), and one has $((v))\leqslant \|v(y)\|< c$ by Prop. 6.