Spaces of Continuous Functions, Banach Spaces and Abstract Equations

Abstract

The combination of the structure of a vector space with the structure of a metric space naturally produces the structure of a normed space and a Banach space, i.e., of a complete linear normed space. The abstract definition of a linear normed space first appears around 1920 in the works of Stefan Banach (1892–1945), Hans Hahn (1879–1934) and Norbert Wiener (1894–1964). In fact, it is in these years that the Polish school around Banach discovered the principles and laid the foundation of what we now call linear functional analysis. Here we shall restrain ourselves to introducing some definitions and illustrating some basic facts in Sections 9.1 and 9.4.