I have $X$ a (non-reflexive) Banach space and $B\subset X$ a weakly closed convex subset.
I wonder under what additional conditions (other than weak compactness) $B$ remains weakly-star closed in $X^{**}$.
My take on this:- the canonical embedding $J:(X,w)\to (X^{**},w^*)$ is linear continuous and my question refers to finding on what sets $B$ is $J$ a closed map.
Another remark: - because $B$ is strongly closed in $X$ it is so in $X^{**}$. Combined with $B$ convex that yields that $B$ is weakly closed in $X^{**}$. It remains to get to the weak-star topology on $X^{**}$.