Let $X$ be a Banach space, denote the closed unit ball of $X$ by $D_1$ and consider a locally convex Hausdorff topology $\tau$ on $X$ that is coarser than the Banach topology, for example $\tau$ could be the weak topology.
Let $V\subseteq X$ be a vector sub-space, is the following equivalence true?
$V$ is closed in $\tau$ $\iff$ $V\cap D_1$ is closed in $\tau$.
If $\tau$ comes from a norm the statement is true, as any $\tau$-convergent sequence must then be bounded wrt the original norm, but for a locally convex topology one cannot guarantee the norm-boundedness of a convergent net and the argument doesn't work so simply.