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.,

coveringcompactness) 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?)