Basic Concepts in Banach Spaces

Most of the theory presented in this text is valid for both real and complex scalar fields. When the proofs are similar, we formulate the theorems without specifying the field over which we are working. When theorems are not valid in both fields or their proofs are different, we specify the scalar field in the formulation of a theorem. K denotes simultaneously the real (R) or complex (C) scalar field. We use N for {1,2,...}. All topologies are assumed to be Hausdorff. In particular, by a compact space we mean a compact Hausdorff space.