Topological Aspects

Abstract

In our introduction we stressed the analogy between Euclidean spaces and Hilbert spaces. This analogy works well as long as only the vector space and the geometric structures of a Hilbert space are concerned. But in the case of infinite dimensional Hilbert spaces there are essential differences when we look at topological structures on these spaces. It turns out that in an infinite dimensional Hilbert space the unit ball is not compact (with respect to the natural or norm topology) with the consequence that in such a case there are very few compact sets of interest for analysis. Accordingly a weaker topology in which the closed unit ball is compact is of great importance. This topology, called the weak topology, is studied in the second section to the extent needed in later chapters.