WOT convergence in the unit ball of B(X)

Question

My questions is (probably) related to:

On separable Hilbert space $H$, weak operator topology is metrizable on bounded parts of $B(H)$

1. Does the theorem quoted in the above question, together with the fact that the unit ball of $B(H)$ (or of any $B(X)$ where $X$ is reflexive) is WOT compact, imply that any sequnce in the unit ball of $B(H)$ has a convergent subsequence? In other words, is the unit ball sequentially-WOT compact?

2. Since WOT and SOT coincide on convex sets, does this mean that any sequence in the unit ball of $B(H)$ has a SOT-convergent subsequence?

3. Do the above (if indeed true) hold when $X$ is a separable reflexive Banach space? Does the proof about metrizability of WOT on the unit ball of $B(H)$ hold in $B(X)$ as well?

Answer 1

1. Yes if and only if $H$ is separable.
2. Yes if and only if $H$ is separable.
3. If I remember correctly yes. Please check this survey paper.