Definition 4

Tag 07L1, permanent. Cite it as [Bourbaki, Tag 07L1].

Topological Vector Spaces, chapter III, § 1. Bornology in a topological vector space

Let $E$ be a locally convex space. A bornology $\mathscr{B}$ on $E$ is said to be adapted to $E$ , if it is convex, is composed of bounded subsets of $E$ and if the closure of every set of $\mathscr{B}$ belongs to $\mathscr{B}$ .

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

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