ABSTRACT
Functional analysis appears as a rather complex blend of algebra and topology, and it should therefore surprise no one that the development of these two branches of mathematics had a strong influence on its own evolution. As a matter of fact, it is almost impossible to dissociate the early history of general topology from the beginnings of functional analysis, since the sets and spaces which attracted most attention consisted of functions. Spectraltheory and dualitytheory are closely connected with what was the main motivation for the creation of Functional analysis, and remains to this day its most important testing ground for its applications, the theory of partial differential equations.
