Proposition 1

Let $E$ be a locally convex space. The set of bounded subsets of $E$ is an adapted bornology.

We need to establish the following properties : a) If $B$ is a bounded subset of $E$, every subset of $B$ is bounded. b) The union of two bounded subsets is bounded. c) Every set that is homothetic to a bounded set is bounded. d) The closed convex balanced envelope (II, p. 13) of a bounded subset is bounded. If $p$ is a continuous semi-norm on $E$, the balls of $p$ are convex, balanced, closed and the set homothetic to a ball is a ball. Hence, if $p$ is bounded on two subsets $X$ and $Y$ of $E$, it is also bounded on the closed convex balanced envelope of $X \cup Y$, and on the sets homothetic to these. This establishes properties $b), c)$ and d), and a) is obvious.