Baggett L.W. Functional Analysis: A Primer

Boca Raton: CRC Press, 1991. — 274 p. — ISBN 0824785983.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. I assume that the students studying from this book have completed a course in general measure theory, so that terms such as outer measure, σ-algebra, measurability, Lp spaces, product measures, etc. should be familiar. In addition, I freely use concepts such as separability and completeness from metric space theory (making particular use of the Baire category theorem at several points in Chapter IV), and I employ the general Stone- Weierstrass theorem on several occasions. I also think that many aspects of general topology were in fact invented to support the concepts in functional analysis, and I draw on these results in some rather deep ways. Thinking that those aspects of general topology that are most critical to this subject, e.g., product topologies, weak topologies, convergence of nets, etc., may not be covered in sufficient detail in many elementary topology courses, I go to some effort to explain these notions carefully throughout the text. I do not intend to include here the most general cases of theorems and definitions, believing that my versions are both hard enough and deep enough for a student’s first go at this subject. For example, I consider only locally compact topological spaces that are second countable, measures that are σ-finite, and Hilbert spaces that are separable.
The second half of the book centers on the Spectral Theorem in Hilbert space, the most important theorem[/b] of functional analysis in my view.Preliminaries.
The Riesz representation theoreм.
Topological vector spaces and continuous linear functionals.
Normed linear spaces and Banach spaces.
Dual spaces.
Applications to analysis.
Axioms for a mathematical model of experimental science.
Hilbert spaces.
Projection-valued measures.
The spectral theorem of Gelfand.
Applications of spectral theory.
Nonlinear functional analysis, infinite-dimensional calculus.True PDF (A5 format)