ABSTRACT
This chapter discusses the study of Banach spaces. The quest for compactness stems from the fundamental fact of life that compactness is usually the only means by which people can establish the existence of certain mathematical objects via a limiting process, typically because no explicit construction of them is possible whatsoever. Sequential compactness is more useful than compactness itself, since it allows people to construct algorithmically an approximation in countably-many steps. Therefore, it allows them to prove that a limiting object exists, by merely constructing a bounded sequence of "confined objects", or, as they typically say, by obtaining an a priori uniform estimate. Along the way people shall also explore new and fascinating structures which emerge from the weakening of the usual norm topologies. The chapter defines the notion of weak topology on a normed space. It concludes with the observation regarding weak convergence on a Banach space which is the dual space of some other space.
