The marriage of algebra and topology has produced many beautiful and intricate subjects in mathematics, of which perhaps the broadest is functional analysis. My aim has been to write a textbook with which graduate students can master at least some of the powerful tools of this
subject. Because I think that one learns best by doing, I believe that it is critical that the students using this book in a course work the exercises.
As an integral part of the book, they have been designed to provide practice in mimicking the techniques that are presented here in the proofs, as well as to lead the novice through fairly elaborate arguments that establish important additional results. The instructor is encouraged and expected to add theorems and examples from his or her own experiences and preferences, for I have quite deliberately restricted this presentation according to my own. My style is to state relatively few theorems, each having a fairly substantial proof, rather than to present a long series of lemmas. The student should read these substantial proofs with pencil in hand, making sure how each step follows from the previous ones and filling in any details that have been left to the reader.
I propose this text for a one-year course. The first six chapters constitute a general study of topological vector spaces, Banach spaces, duality, convexity, etc., concluding with a chapter that contains a number of applications to classical analysis, e.g., convolution, Green's functions, the Fourier transform, and the Hilbert transform.