Proposition 2

Tag 07L4, permanent. Cite it as [Bourbaki, Tag 07L4].

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

In a locally convex space $E$ , every precompact set is bounded. Let $A$ be a precompact subset of $E$ , and $V$ be a convex balanced neighbourhood of $0$ . There exists a finite sequence $(a_i)_{1 \leq i \leq n}$ of points of $A$ such that

A \subset \bigcup_{1 \leq i \leq n} (a_i + V).

Since

B = \{a_1, ..., a_n\}

is bounded, there exists a scalar

\lambda

such that

0 < \lambda < 1

and

\lambda B \subset V

; we have

\lambda A \subset \lambda B + \lambda V \subset V + V

, from which the proposition follows.

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

Cited by

Nothing.

Cites

Nothing.