Definition 2

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

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

Let $E$ be a vector space. A bornology $\mathscr{B}$ on $E$ is said to be convex, if for every $X \in \mathscr{B}$ and $t \in K$ , the homothetic set $tX$ and the convex balanced envelope $\Gamma(X)$ (II, p. 10) of $X$ belong to $\mathscr{B}$ .

If $X$ and $Y$ are two subsets of $E$ , we have

X + Y \subset 2\Gamma(X \cup Y)

\lambda X \subset t\Gamma(X) \quad \text{for} \quad |\lambda| \leq t .

Consequently, if $\mathscr{B}$ is a convex bornology on $E$ , if $A$ is a bounded subset of $K$ and if $X, Y$ belong to $\mathscr{B}$ , then $X + Y \in \mathscr{B}$ and $A.X \in \mathscr{B}$ .

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

Cited by

Nothing.

Cites

II, p. 10