Is the unit ball in $X''$ weak-star closed?
I reached a point in my argument where it would suffice to show that the unit ball in $X''$ is weak-star closed. (Where $X$ is just some topological space and $X'$ is the continuous dual of $X$, etc.)
If $X$ was reflexive then of course this result would hold due to the unit ball in $X$ being weakly closed, therefore the unit ball in $X''$ would be weak-star closed.
But does this result hold in general?