Topological Aspects

Abstract

First, we recall the concept of compactness and proceed to prove that the closed unit ball of a normed space is compact if, and only if, this space is finite dimensional (Theorem of Riesz). Next, the weak topology is defined on a Hilbert space and its basic properties are explored. In particular, we show when a weakly convergent sequence converges strongly (in norm). We conclude this chapter by proving the Banach–Steinhaus theorem (uniform boundedness principle) for families of continuous linear functionals on Banach spaces and the Banach-Saks theorem.