Locally convex spaces

Question

Suppose we assume the existence of a weakly compact set with non-empty interior in a LCS X. Does imply that X is a reflexive locally convex space? would it be Banach?

Answer 1

Yes (if the space is Hausdorff). First observe that $X$ is normed since there is a bounded open set (and hence a bounded absolutely convex $0$-neighbourhood whose Minkowski-functional is then a norm which induces the given locally convex topology). Moreover, the unit ball of this norm is relatively weakly compact and weakly closed. Hence it is weakly compact and this is equivalent to reflexivity.