Closed convex hull of a compact set ($\overline{\mathrm{conv}}K$)

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