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Building on the success of the two previous editions, Introduction to Hilbert Spaces with Applications, 3E, offers an overview of the basic ideas and results of Hilbert space theory and functional analysis. It acquaints students with the Lebesgue integral, and includes an enhanced presentation of results and proofs. Students and researchers will benefit from the wealth of revised examples in new, diverse applications as they apply to optimization, variational and control problems, and problems in approximation theory, nonlinear instability, and bifurcation. The text also includes a popular chapter on wavelets that has been completely updated. Students and researchers agree that this is the definitive text on Hilbert Space theory.
* Updated chapter on wavelets
* Improved presentation on results and proof
* Revised examples and updated applications
* Completely updated list of references .
- Sales Rank: #945434 in Books
- Published on: 2005-10-13
- Ingredients: Example Ingredients
- Original language: English
- Number of items: 1
- Dimensions: 9.02" h x 1.50" w x 5.98" l, 2.35 pounds
- Binding: Hardcover
- 600 pages
Review
"...this is a very useful and good book and it can find a place in the library of anybody interested in functional analysis, particularly Hilbert Spaces and their applications."
-MAA REVIEWS
From the Back Cover
The Second Edition if this successful text offers a systematic exposition of the basic ideas and results of Hilbert space theory and functional analysis. It includes a simple introduction to the Lebesgue integral and a new chapter on wavelets. The book provides the reader with revised examples and updated diverse applications to differential and integral equations with clear explanations of these methods as applied to optimization, variational and control problems, and problems in approximation theory, nonlinear instability, and bifurcation.
Key Features and Enhancements in the New Edition
* Systematic exposition of the basic ideas and results of Hilbert space theory
* Introduction to the Lebesgue integral
* New chapter on wavelets
* Improved presentation on results and proof
* Revised examples and updated applications
* Completely updated list of references
About the Author
Lokenath Debnath is Professor of the Department of Mathematics and Professor of Mechanical and Aerospace Engineering at the University of Central Florida in Orlando. He received his M.Sc. and Ph.D. degrees in pure mathematics from the University of Calcutta, and obtained D.I.C. and Ph.D. degrees in applied mathematics from the Imperial College of Science and Technology, University of London. He was a Senior Research Fellow at the University of Cambridge and has had visiting appointments to several universities in the United States and abroad. His many honors and awards include two Senior Fulbright Fellowships and an NSF Scientist award to visit India for lectures and research. Dr. Debnath is author or co-author of several books and research papers in pure and applied mathematics, and serves on several editorial boards for scientific journals. He is the current and founding Managing Editor of the International Journal of Mathematics and Mathematical Sciences.
Piotr Mikusinski received his Ph.D. in mathematics from the Institute of Mathematics of the Polish Academy of Sciences. In 1983, he became visiting lecturer at the University of California at Santa Barbara, where he spent two years. He is currently a member of the faculty in the Department of Mathematics at the University of Central Florida in Orlando. His main research interests are the theory of generalized functions and real analysis. He has published many research articles and is co-author with his father, Jan Mikusinski, of An Introduction to Analysis: From Number to Integral.
Most helpful customer reviews
26 of 28 people found the following review helpful.
Good book to teach yourself this interesting subject
By A Customer
I'm a statistician who has been using Part 1 of this book to teach myself the basics of Hilbert space theory. So far, I've been very pleased with it.
I've only run into one argument that assumed a fact that wasn't made fairly plain earlier in the development (for Corollary 4.6.1, I had to resort to Rudin's Functional Analysis text to learn why everywhere-defined positive operators on Hilbert spaces are bounded). Functional analysis seems to be a subject where you'll want to have a few different texts on hand in case what one author considers obvious is not so obvious to you!
Nice features of this book include
--an interesting proof of the Banach-Steinhaus theorem that uses a clever Diagonalization Theorem instead of the Baire Category theorem
--an entire chapter introducing the Lebesgue integral and developing its properties without auxiliary concepts such as measure: I found this chapter to be an interesting alternative way to look at the Lebesgue integral. My only quibble with it is that it quotes a version of Fatou's lemma that only applies to functions with limits (almost everywhere). In probability theory, Fatou's lemma is often applied on liminf's and limsup's of functions that don't have limits
--including the Lebesque integral chapter, a total of four solid chapters that develop the theory systematically and clearly enough for careful readers to follow. These comprise Part 1, which I'm almost finished with.
--five chapters with applications. I've only skimmed these, but together they really make this book seem like a terrific value. There's a chapter on applications to integral and differential equations, one on generalized functions and PDEs (e.g. distribution theory), a really interesting looking chapter on Quantum Mechanics, a chapter on wavelets that includes a terrific and concise section with historical remarks and a chapter on optimization problems, including the Frechet and Gateaux differentials, which comprise one of my major motivations for reading this book
--answers to selected exercises (HOORAY!)
This book can be used as the primary text for people who want to acquire a good understanding of Hilbert space theory so that they can use it to solve applied problems: at least, that's how I'm trying to use it! This book is a good value for scientists and engineers.
18 of 21 people found the following review helpful.
Very good book
By Raymond Jensen
Lokenath Debnath, like many authors from India, I am finding, write solid mathematical texts. These texts tend to be well-organized, clear, and do not leave out or fail to emphasize important concepts. The proofs are easy to understand. It does not take a week just to read a few pages.
This book by Debnath, is a good example of a book fitting the above criteria. It is an excellent book for self-study of Hilbert spaces, Fourier Transforms and other subjects in Functional Analysis. I found it to be a useful supplement to Folland's "Real Analysis" which I used as a 1st-year graduate student in mathematics. In fact, this book saved me a few times, when I had to figure out solutions to difficult homework excercises. One example comes to mind is a homework assignment (I think that it was out of Folland's book) involving Rademacher and Walsh functions, which are covered in this book. I also found this text for useful in studying for my candidacy examination.
In summary, this book is would make an excellent addition to your library. (If you are also interested in the subject of elliptic functions, then "Elliptic and Associated Functions with Applications" by Debnath and M. Dutta (World Press Private Ltd., Calcutta, 1965), may interest you. It is, like the above text, excellent, but very difficult to find!)
0 of 0 people found the following review helpful.
Five Stars
By mundaka
Clearest introduction I know to inner-product spaces.
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