Let $C$ is a convex subset in a normed linear space, does $\mbox{Int} (C) = \mbox{Int} \left(\overline{C}\right)$ while $\mbox{Int} (C) = \emptyset$?

Question

Let $C$ be a convex subset in normed linear space $X$. It is known that $$\mbox{Int} (C) = \mbox{Int} \left(\overline{C}\right)$$ if $\mbox{Int} (C)$ is nonempty. How it would be if $\mbox{Int} (C)$ is empty? If the above formula is not true, is there any counterexample?

Answer 1

Any proper dense linear subspace $M$ is a counter-example. Note that $Int M=\emptyset$ and $Int\overline{M}$ is the whole space.

Specific example: The set of all finitely non-zero sequences in $\ell^{p}$.