Corollary 1

Tag 07U3, permanent. Cite it as [Bourbaki, Tag 07U3].

Topological Vector Spaces, IV, p. 25, § 3. Dual of a Fréchet space

Let $E$ be a locally convex metrizable space. Every precompact subset of $E$ is contained in the closed convex balanced envelope of the set of points of a sequence converging to 0.

This follows from the fact that the topologies $\mathscr{T}_{pc}$ and $\mathscr{T}_{\mathfrak{H}}$ are identical, on account of prop. 2 of III, p. 15.

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

Cited by

Nothing.

Cites

III, p. 15