Let $X$ be a real Banach space. Apart from the norm topology, we can consider the following weak topologies on $X$:
- the weak toplogy, defined as the initial topology with respect to $X^*$. In other words, it is the coarsest topology for which all $f\in X^*$ are continuous.
- the weak sequential topology, which is essentially the topology induced by weak convergence. More precisely, we call a set closed if it is weakly sequentially closed, and this induces a topology.
It is easy to see that the weak topology is the weaker of the two (since every $f\in X^*$ is weakly sequentially continuous). Moreover, it is well known that a weakly sequentially closed set is not necessarily weakly closed. However, the picture is not so clear when it comes to compactness:
By the Eberlein-Smulian theorem, weak compactness coincides with weak sequential compactness. However, it is important to note that weak sequential compactness means sequential compactness, not compactness in the weak sequential topology (!!). In particular, this raises the following question:
What does ordinary compactness (i.e., covering compactness) look like in the weak sequential topology? Is it equivalent to weak (sequential) compactness?
(Also: if these are not equivalent in general, what about the case of convex sets?)