Let $X$ a topological vector space and $A\subseteq X$ a subspace. Let $co(A)$ the convex hull of $A$ (the smallest convex subspace containing $A$) and $\overline{co}(A)$ the closed convex hull of $A$ (the smallest closed and convex subspace containing $A$).
What is the relationship between $\overline{co}(A)$, $\overline{co(A)}$ and $co(\bar{A})$? Are they always equal?
In a related question closure, convex hull and closed convex hull they found different counterexamples, but in the book "Topological Vector Spaces" by Lawrence Narici and Edward Beckenstein they give a proof of the fact that they are all equal:
