If $X$ is a dual space, the ball $B_X$ is weak star compact. Is the sphere generally weak star compact?
I think the answer is no because in $\ell^2$, the sequence $(e_n)_n$ converges weakly to zero, and so no subsequence converges weakly to an element with norm one.
The context for the question is the following statement from Wikipedia: Let $X$ be a $C^*$ algebra and with unit. The set of states (continuous linear functionals $f$ on $X$ such that $||f||=1$ and $f(x)\geq 0$ when $x\geq 0$) is a weakly compact set.
Best Answer
You are correct.
For a reflexive space $X$, it is not difficult to prove that the weak closure of the unit sphere $S_{X^*}$ is the closed unit ball $B_{X^*}$ (you can read the proof here). Weak and weak* topology coincide on the dual space of a reflexive space.