ABSTRACT
This chapter discusses "weakened" topologies on Banach spaces, with our end goal being the quest for (sequential) compactness results for these topologies. For reflexive Banach spaces, namely for those which coincide with their bidual space, weak compactness of closed bounded sets holds indeed true. For dual Banach spaces, there exists an even weaker topology, known as the weak* topology which indeed has the desired compactness properties. For general Banach spaces, it is usually very difficult to check directly from the definition whether a space is reflexive or not. The chapter concludes the study of weak topologies on Banach spaces by introducing a geometric criterion which is much easier to check and shows that it is a sufficient condition for reflexivity.
