# Convex Hull of Precompact Subset is Precompact

I'm trying to prove that, if $K$ is a precompact (I've also heard the phrase totally bounded used for this) subset of a Banach Space $X$, then its convex hull is also precompact.

I've come across a similar statement with Hilbert spaces that suggested I fix some $x\in X$ and define a bounded conjugate linear form on $X$ by $x\mapsto B(x,y)$ and then use the Riesz Representation Theorem. I'm not sure if this can be generalized to a proof for general Banach spaces. Any help would be appreciated.

Thank you.

A subset $$K$$ of an abelian topological group $$X$$ is called precompact provided for each neighborhood $$U$$ of the zero of the group $$X$$ there exists a finite subset $$F$$ of the space $$X$$ such that $$F+U\supset K$$.
It seems the claim holds for each locally convex topological vector space $$X$$. Indeed, let $$U$$ be an arbitrary neighborhood of the zero of the group $$X$$. Since the space $$X$$ is locally convex, there exists a convex open neighborhood $$V$$ of the zero of the space $$X$$ such that $$V+V\subset U$$. Since the set $$K$$ is precompact, there exists a finite subset $$F$$ of the space $$X$$ such that $$F+V\supset K$$.
Let $$x$$ be a point of the convex hull $$\operatorname{conv} K$$ of the set $$K$$. Then there exist a natural number $$n$$, non-negative real numbers $$\lambda_1, \dots, \lambda_n$$ with sum $$1$$, and points $$x_1,\dots,x_n$$ of the set $$K$$ such that $$x=\sum_{i=1}^n \lambda_ix_i$$. Since $$F+V\supset K$$, for each $$i$$ from $$1$$ to $$n$$ there exist points $$f_i\in F$$ and $$y_i\in V$$ such that $$x_i=f_i+y_i$$.
Then $$x=\sum_{i=1}^n \lambda_i (f_i+y_i)= \sum_{i=1}^n \lambda_i f_i+\sum_{i=1}^n \lambda_i y_i$$. The second summand is contained in the set $$V$$, because the set $$V$$ is convex. The first summand is contained in the convex hull $$\operatorname{conv} F$$ of the finite set $$F$$.
The set $$\operatorname{conv} F$$ is compact as a continuous image of a compact set $$\Lambda_n\times F^n$$, where $$\Lambda_n=\{\lambda\in \mathbb R^n: \sum_{i=1}^n \lambda_i=1$$ and $$\lambda_i\ge 0$$ for every $$i\}$$ is an $$n-1$$-dimensional simplex. Since $$\{z+V: z\in \operatorname{conv} F \}$$ is an open cover of the compact set $$\operatorname{conv} F$$, there exists a finite a finite subset $$F’$$ of the set $$\operatorname{conv} F$$ such that $$\bigcup \{z+V: x\in F’\}\supset F$$. Finally we obtain that $$\operatorname{conv} K\subset \operatorname{conv} F+V\subset F’+V+V\subset F'+U$$.