I have only recently been introduced to the notions of weak and weak-* topologies. I know the definition of both as well as an idea of what they represent.
It is clear through Banach-Alaoglu and Helly where
Banach Alaoglu states $\overline{B}^{*}_{1}$ is weak$-*$ compact where $\overline{B}^{*}_{1}$ is the closed unit ball in the dual of normed space $X$.
while Helly goes further that if $X$ is also separable then $\overline{B}^{*}_{1}$ is weak$-*$ sequentially compact.
Taking both theorems into account, it is clear that weak$-*$ sequential compactness is not equivalent to weak$-*$ compactness
But I thought $X$ was not metrizable only in the weak topology, nothing was said about the weak$-*$ topology.
Further, are weak$-*$ sequential continuity and weak$-*$ continuity equivalent since we are no longer in metric spaces? As well as, are weak sequential continuity and weak continuity equivalent? Thanks for any clarification.