Proposition 3

— In a locally convex space E, the balanced convex envelope of a precompact set is itself a precompact set.

Let A be a precompact set in E. Given V, a balanced convex neighbourhood of 0 in E, there exist finitely many points $a_i \in A$ ($1 \leq i \leq n$) such that A is contained in S, the union of the neighbourhoods $a_i + V$ ($1 \leq i \leq n$). Thus C, the balanced convex envelope of A, is contained in T the balanced convex envelope of S; but T is contained in B + V, where B denotes the convex envelope of the finite set of points $a_i, -a_i$ ($1 \leq i \leq n$). Now B is precompact (II, p. 14, cor. 2); hence there exist finitely many points $b_k \in B$ ($1 \leq k \leq m$) such that $B_k$ is contained in the union of the neighbourhoods $b_k + V$. Then C is contained in the union of the neighbourhoods $b_k + 2V$, and the proposition is proved.

Note that, in an infinite dimensional locally convex Hausdorff space, the convex envelope of a compact set is not necessarily closed (II, p. 74, exerc. 3).