I am reading a statement of Goldstein's Theorem that reads (here $J$ is the embedding of $X$ into $X^{**}$):
Let $X$ be a normed linear space, $B$ the closed unit ball of $X$, and $B^{**}$ the closed unit ball of $X^{**}$. Then the the weak-* closure of $J(B)$ is $B^{**}$.
A statement in the proof reads:
We leave it as an exercise to show $B^{**}$ is weak-* closed.
I'm am confused by this statement, we assumed $B^{**}$ to be the closed unit ball in $X^{**}$. Doesn't this mean $B^{**}$ is closed with respect to the topology on $X^{**}$ (which $\textit{is}$ the weak-* topology)?
When we say "the closed unit ball in $X^{**}$" without any specification of the topology, we usually mean the closure in the norm of $X^{**}$. In other words "the topology on $X^{**}$" is that inudced by the norm, not the weak-star topology.