Let $E$ be a real topological vector space and $C\subset E$ be a convex subset. Is this true that $\mathrm{Int}(\bar{C})=\mathrm{Int}(C)$? Here $\bar{C}$ is the closure of $C\subset E$ and $\mathrm{Int}$ is the set of interior points.
We know that the statement holds for finite dimensional $E$. cf. Why does a convex set have the same interior points as its closure?