This is a book for people who want to use functional analysis to justify approximate methods in mechanics and inverse problems. It aims to provide such researchers with the tools they need without having to assimilate or skip through concepts they do not need. For the applied mathematician, the essential difficulty attending the study of functional analysis is that the pure mathematicians who have developed the field have carried the process of abstraction, which is the essence of functional analysis, to increasingly higher levels. In this work, the authors have kept the level of abstraction high enough for the majority of applications, and have resisted the temptation to abstract to the limit. It can be used for a lecture course or for self study; there are exercises throughout, and references for further study. This new edition corrects a number of typographical and logical errors in the first edition.