- Tapa blanda: 390 páginas
- Editor: Dover Publications Inc.; Edición: Dover Ed (28 de marzo de 2003)
- Colección: Dover Books on Mathematics
- Idioma: Inglés
- ISBN-10: 0486422585
- ISBN-13: 978-0486422589
- Valoración media de los clientes: Sé el primero en opinar sobre este producto
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Applied Functional Analysis (Dover Books on Mathematics) (Inglés) Tapa blanda – 28 mar 2003
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This introductory text examines many important applications of functional analysis to mechanics, fluid mechanics, diffusive growth, and approximation. Discusses distribution theory, Green's functions, Banach spaces, Hilbert space, spectral theory, and variational techniques. Also outlines the ideas behind Frechet calculus, stability and bifurcation theory, and Sobolev spaces. 1985 edition. Includes 25 figures and 9 appendices. Supplementary problems. Indexes.
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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.
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.