5.1 Baire Category Theorem

In this chapter, we explore the theory of bounded linear operators. As we will see, the completeness of the underlying spaces plays an important role. We have already seen one example of this in Theorem 2.3.13, where we showed that B(E, F) is a Banach space whenever F is complete. Here, that simple minded use of completeness gives way to a deeper understanding with the introduction of the Baire Category Theorem. This innocuous-looking result allows us to prove two results that are fundamental to the subject: the Principle of Uniform Boundedness (Theorem 5.2.1) and the Open Mapping Theorem (Theorem 5.3.4). These two theorems and their many applications are the main focus of this chapter.