Banach Spaces

This chapter explores the properties of operators and functionals on general Banach spaces, with the aim of generalizing various results on Hilbert spaces. The main theorems in this regard, namely the Open Mapping theorem, the Hahn-Banach theorem, and the Uniform Boundedness theorem, are proved. These resolve a number of issues in the theory of Banach spaces: complementary subspaces, compact and Fredholm operators, the separating hyperplane theorem, the generalization of orthogonal subspaces (annihilators) and adjoint operators, and the closed range theorem. The chapter concludes with a section on strong and weak convergence.