Let $X$ be an infinite dimensional Banach space, and $C\subset X$ a convex set. Is it always true $\text{ri}(C)=\text{ri}\left(\overline{C}\right)$?
What about for general topological vector spaces? Thanks.
Background: I've been trying to seriously understand results about convex sets in functional analysis. Previously, I've always just take things like Krein-Milman for granted. So initially I'm trying to see whether we always have $\text{ri}(C)\neq \emptyset$. But there's a counterexample here. Then I guessed maybe that's true when $C$ is closed? Then, I can see the same counterexample still stands when taking the closure. Another counterexample for closed convex set is here. These counterexamples make me wonder whether the statement above is true.
No. In an infinite-dimensional Banach space $X$, you can have a proper subspace $E$ that is dense. $E$ is convex, and $\text{ri}(E) = E$, but $\text{ri}(\overline{E}) = \overline{E} = X$.