Corollary 1

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

Topological Vector Spaces, chapter III, § 3. Spaces of continuous linear mappings

— For an equicontinuous subset H of $\mathscr{L}(E; F)$ to be relatively compact in $\mathscr{L}_s(E; F)$ , it is necessary and sufficient that for all $x \in E$ , the set $H(x)$ of all $u(x)$ as $u$ ranges over H, is relatively compact in F.

In fact, this condition is necessary and sufficient for $\overline{H}$ to be compact in $F^E$ (GT, I, § 9, No. 5, cor.).

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

Cited by

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Cites

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