An Introduction to Functional Analysis

Once again we use the Baire Category Theorem to prove results about linear maps between Banach spaces. We prove the Open Mapping Theorem and, as a corollary, the Inverse Mapping Theorem, which allows for some simplification in the spectral theory of bounded operators. As an application, we prove the existence of a ‘basis constant’ for any Schauder basis in a separable Banach space. Finally, we use the Inverse Mapping Theorem to prove the Closed Graph Theorem, which gives an alternative way to check whether a linear map T from X into Y is bounded, provided both X and Y are Banach spaces.