= Closed convex hull of a compact set
{title2=$\overline{\operatorname{conv}}K$}
In a <Banach space>, the <closed convex hull of a compact set> is compact in the <norm topology>. Given $\varepsilon>0$, choose a finite $\varepsilon$-net $\{y_i\}$ for $K$. Every <convex combination> of points of $K$ is within $\varepsilon$ in <norm> of $\operatorname{conv}\{y_i\}$, by replacing its terms by their approximating $y_i$. This finite-dimensional compact <convex set> has a finite $\varepsilon$-net, giving a finite $2\varepsilon$-net for $\operatorname{conv}K$. Its closure is therefore <totally bounded> and <complete>, hence compact. <Completeness> of the ambient space is used at this last step.
Back to article page
