Introduction to Hilbert Spaces with Applications

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!)


Good book to teach yourself this interesting subject   August 11, 2003
 8 out of 9 found this review helpful

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.


Great and Clear   April 20, 2000
 9 out of 11 found this review helpful

Debhath and Mikusinski used great and clear Mathematics and diagrams to explain the theory and applications. I especially like chapter seven "Mathematical Foundations of Quantum Mechanics" and chapter eight "Wavelet". This book is suitable for graduate engineering students.