The convex hull of a precompact set in a Banach space is precompact

Question

At the begining of the Chapter 21 of Peter Lax's functional analysis, he says that

If C is a precompact set in a Banach space, so is its convex hull.

can be deduced from the following properties:

(a) S is precompact if and only if every sequence of points of S contains a Cauchy subsequence.

(b) S is precompact iff for every $\epsilon >0 $ it can be covered by a finite number of balls of radius $\epsilon$.

I can not see how we can deduce it from these properties, can someone help me? Thanks!

Answer 1

Let $n_k$ be a finite $\epsilon$-net for $C$ and let $A= \operatorname{co} C$.

Then $N=\operatorname{co} \{n_k \}_k$ and note that $N$ is contained in a finite dimensional subspace and hence is compact. In particular, $N$ has a finite $\epsilon$-net $\nu_k$.

I claim that this is an $2\epsilon$-net for $A$.

If $x \in A$, then $x = \sum_i \lambda_i c_i$ with $\sum_i \lambda_i = 1$, $\lambda_i \ge 0$, and $c_i \in C$.

Choose $n_{k_i}$ to be an element of $C$'s $\epsilon$-net nearest $c_i$, then $n= \sum_k \lambda_i n_{k_i}$ satisfies $\|x-n\| < \epsilon$. Now choose $\nu_k$ such that $\|n-\nu_k\| < \epsilon$, then $\|x-\nu_k\| < 2 \epsilon$.