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Download Applied Functional Analysis (Dover Books on Mathematics) epub

by D.H. Griffel

A stimulating introductory text, this volume examines many important applications of functional analysis to mechanics, fluid mechanics, diffusive growth, and approximation. Detailed enough to impart a thorough understanding, the text is also sufficiently straightforward for those unfamiliar with abstract analysis. Its four-part treatment begins with distribution theory and discussions of Green's functions. Essentially independent of the preceding material, the second and third parts deal with Banach spaces, Hilbert space, spectral theory, and variational techniques. The final part outlines the ideas behind Frechet calculus, stability and bifurcation theory, and Sobolev spaces. 25 Figures. 9 Appendices. Supplementary Problems. Indexes.
Download Applied Functional Analysis (Dover Books on Mathematics) epub
ISBN: 0486422585
ISBN13: 978-0486422589
Category: Science
Subcategory: Mathematics
Author: D.H. Griffel
Language: English
Publisher: Dover Publications; Reprint edition (June 14, 2002)
Pages: 400 pages
ePUB size: 1370 kb
FB2 size: 1753 kb
Rating: 4.5
Votes: 518
Other Formats: docx azw txt rtf

I don't know what all these people are trying to do by saying that this book is "the best book on functional analysis". It's not. I think if one is looking for a "Functional Analysis-Lite" kind of book, you could do better. If you want to learn about Banach and Hilbert spaces, other books would be more thorough and helpful, especially for physicists learning about it grad school having to grapple with it all the time.

The book is divided into 4 parts, and I will discuss each part.

I. Distribution Theory and Green's Functions
II. Banach Spaces and Fixed Point Theorems
III. Operators in Hilbert Spaces
IV. Further Developments

PART I: This is actually a bit more confusing and unclear than it needs to be. A lot of it could be done with more motivation. The actual chapter on Green's functions and PDEs is pretty standard, more or less. I've seen Green's functions discussed better in books by Partial Differential Equations of Mathematical Physics and Integral Equations and also in Methods of Theoretical Physics, Part I. For PDEs and Green's functions solutions, I would recommend a book devoted to PDEs that also covers Fourier Transform methods - there are plenty.

Part II: The chapter on Normed Spaces is not too bad. The discussion is helpful. It helps one understand only the very basics. If you are an applied mathematician developing theories for numerics, or trying to solve intricate PDE problems, I recommend looking at the first half of Elements of the Theory of Functions and Functional Analysis and especially Introductory Functional Analysis with Applications. Their discussion of the fixed point theorems and the Contraction Mapping Theorem (of which Newton's method of solving for zeros is an example), IMHO, are far superior and enlightening. The two just mentioned actually teach you how to think as a mathematician, relatively painlessly. The only advantage is that Griffel's book touches on some modern applications which you may or may not encounter.

Part III: This portion is like a physicists introduction to Hilbert spaces and applications. Here the last book mentioned does a superb job at introducing the material - both Hilbert spaces and Operator theory. In fact, there is also a chapter on on unbounded operators in quantum mechanics in the Kreyszig book that I noticed is missing in Griffel's. The advantage of Griffel's book is that there is a pretty good discussion of Variational methods that I've only seen elsewhere in Linear Algebra and PDE books.

Part IV: I actually cannot comment on this because I did not go too deeply into this material. For Sobolev spaces I looked elsewhere. I never needed the Frechet derivative. Other reviews seem to like it, though.

I have heard some of my students use this book for a class (not taught by me). They were thoroughly confused when going over the material in part I. They did not have knowledge of Real Analysis coming in. For students and classes with students like these, I recommend Introductory Functional Analysis with Applications. For students with little to no prior experience with Real Analysis, I recommend Elements of the Theory of Functions and Functional Analysis. For students with experience in Real Analysis, I recommend the last book mentioned, or one of my favorites, Elementary Functional Analysis. This last book covers a lot of the material that Griffel does (not all); it goes deeper into issues regarding normed vector spaces, Hilbert spaces, etc... ; it teaches one to think like a mathematician (applied or otherwise) and is useful, in my opinion, for physicists as well.

I am not an analyst, I taught myself a lot of this material and Griffel was not helpful when I was taking courses that covered the same stuff and when I was trying to learn about functional analysis. I offer the above list of books as alternatives to finding good stuff - I'm sure other resources exist.
When studying chemical quantum mechanics some years ago, much reference was made to Hilbert spaces, functionals, adjoint operators and many of the mathematical constructs associated with functional analysis. The majority of books that I had available at that time offered little practical explanation and were overindulged with obscure details, mathematical proofs and issues far beyond what was needed by the beginning quantum chemist. I think that Griffel's book bridged the gap providing enough material to fully grasp the terminology and theorems needed for the study of quantum mechanics while not overwhelming the beginner. It think that this book would be a nice prelude to the one by Byron and Fuller.
Griffel's book is a great introductory functional analysis text. As the title suggests, it is aimed at applied mathematicians rather than theoreticians. In practical terms, it means that Griffel shows how the tools of functional analysis can be applied to differential equations, dynamical systems, and fourier analysis. Griffel gives proofs of most theorems, skipping proofs only when the proof requires a more sophisticated background than this book assumes. The assumed background of the reader is familiarity with calculus, basic differential equations, and some real analysis and linear algebra. A set of appendices cover the needed results from analysis.
The main strength of Griffel's book is its readability. It is one of the most accessible advanced math books I have encountered, comparable to Munkres' "Topology". Griffel explains the intuitions underlying the abstract concepts he presents. He is also careful to point out when he makes a simplification or omission to avoid a difficult or subtle point more suitable to a pure math treatment of the subject. Furthermore, Griffel explains the logic behind his notation, something that is rarely done in math texts. Each chapter concludes with a set of problems. The problems are challenging, but test and expand the reader's understanding of the material. Hints are given for many of the problems.
Overall, this is an excellent resource for the applied mathematician, engineer, or scientist who wants an accessible introduction to functional analysis. Besides, the price of the Dover Edition makes this book a real bargain.
This is the best book in Introductory functional Analysis book I know and I know a lot of them. Why is it so good? The definitions are very well motivated. Then the subtle points are illustrated with examples.Then there are the theorems all well motivated and with simple ,very well explained proofs. Then there are the applications to engineering and physics .All the aplications are well explained. There is no danger of not understanding the application. Then there are the problems with notes on them at the end ( The author offers you the complete solutions for a pittance)Finally the price....You can not beat this book
This book contains one of the best descriptions of the Frechet derivative (functional differentiation) and its applications that I have ever read. This has always been a mystery to me since it is such a fundamentally useful notion, and crops up everywhere in the subject of nonlinear PDE's and numerical analysis. I would recommend this book to any applied mathematician, and especially to engineers, based on Griffel's attention to applications.
For some time I was trying to find a book on functional analysis that wasn't too technical nor too elementary. Even though there are excellent books on this domain none of them suited me. Griffel's book is exactly what I needed. Well structured, with a lot of examples and an effort to communicate the ideas behind the technicalities, makes the study and understanding of the domain fluent.