When is the normal cone to a closed convex set in a locally convex set maximal monotone?

Question

Let $X$ be a locally convex set with the following property:

(P) $\forall C\subset X$ closed convex, the normal cone $N_C$ is maximal monotone as a multi-valued operator from $X$ to its topological dual $X^*$.

Three questions:

Q1 What can one say about $X$ in general?

Q2 Must $X$ be a Banach space?

Q3 Is Q2 true if, in addition, we know that $X$ is barreled?

Thank you.