Applications of the Baire-Hausdorff Theorem

Abstract

The completeness of a B-space (or an F-space) enables us to apply the Baire-Hausdorff theorem in Chapter 0, and we obtain such basic principles in functional analysis as the uniform boundedness theorem, the resonance theorem, the open mapping theorem and the closed graph theorem. These theorems are essentially due to S. Banach [1]. The termwise differentiability of generalized functions is a consequence of the uniform boundedness theorem.