The Weak and Weak Topologies

The topology induced by a norm on a vector space is a very strong topology in the sense that it has many open sets. This has some advantages, especially since a function whose domain is such a space finds it particularly easy to be continuous, but it also has its disadvantages. For example, an infinite-dimensional normed space always has so many open sets that its closed unit ball cannot be compact. Because of this, many familiar facts about finite-dimensional normed spaces that are based on the Heine-Borel property cannot be immediately generalized to the infinite-dimensional case.