Since the early days of Functional Analysis during the first decades of the twentieth century, the theory of Banach spaces has undergone quick and deep development, progressing both horizontally and vertically. The term “horizontal” is used to denote the expansion of the theory’s boundaries and the establishment of connections with other areas. Illustrations of this include Chap. 9 on Nonlinear Analysis, the development of Distribution Theory and its interconnections, the application of Sobolev spaces to the theory of Partial Differential Equations, the exploration of Banach Algebras (specifically C*-algebras), and the applications of the Hahn–Banach Theorem outlined at the beginning of Chap. 3 .