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- Sales Rank: #490346 in Books
- Brand: McGraw Hill
- Published on: 1973-01
- Ingredients: Example Ingredients
- Original language: English
- Number of items: 1
- Binding: Hardcover
- 397 pages
Most helpful customer reviews
3 of 3 people found the following review helpful.
Pure Excellence
By Prof. Dr.habil. Lothar M. Schmitt
This book presents an excellent comprehensive introduction into the subject "Functional Analysis" which enables the reader to pursue further reading into specialized areas of the field. The style is well-balanced: not too much detail such that the reader does not get bored and overwhelmed with technicalities, little facts and examples; but full proofs with enough detail to verify them without additional resources (i.e., other books) except, perhaps, the little piece of paper on the side for checking, e.g., an algebraic identity. In my private opinion as a professional mathematician, this is the best math book ever written, at least among those which I have seen.
22 of 26 people found the following review helpful.
Outstanding
By Fernando Sanz Gamiz
Hardly can I find words to highlight the goodness of this book. As mentioned by other readers ,it provides elegant, direct and powerfool proofs of the three theorems which constitute the cornserstones of functional analysis (Hanh-Banach, Banach-Steinhaus and Open mapping). These theorems are, in addition, studied in their most general context, namely topological vector spaces.
Specially appealing is its treatment of distributions' theory. It is, as far as I know, the only text which start by defining the rigurous topology on the set of test functions and then obtains the convergence and continuity of functionals (distributions) in terms of this topolgy, which is, indeed, the only way to present and gain insight into these concepts and to reach some results such as completness. In doing otherwise one risk definitions can emerge as artificial and rather arbitrary.
It is, without any doubt, a must have book for those with interest in pure mathematics as well as for those who, eventually, realize that the only way to dominate their area is saling through mathematics.
2 of 2 people found the following review helpful.
The proofs are meticulous even when they are short
By Jordan Bell
This is a well-written book that covers an astoundingly large number of ideas. Some of the proofs Rudin gives demand verifications he does not give, but it is apparent to the reader what needs to be checked and if you do check these things you will not find technicalities that Rudin ignored. (Often experienced mathematicians omit parts of proofs they consider standard and in fact if we fully work out the proof we see that what was written is logically out of order, e.g. statements P and then Q are made when in fact Q needs to be established first to prove P, and therefore rightly frustrates a reader.)
The first three chapters are on topological vector spaces generally and locally convex spaces in particular. These structures are not part of the standard graduate course in functional analysis, which deals only with Banach spaces and Hilbert spaces and may give a uselessly specialized proof of the spectral theorem merely for bounded self-adjoint compact operators, while in fact what one genuinely needs the spectral theorem for is unbounded self-adjoint operators (which Rudin gives in Chapter 13). Moreover, it is impossible even to talk rigorously about distributions without the machinery of locally convex spaces and Fréchet spaces; in a course on partial differential equations it is common to avoid talking about what it means to say that a distribution is continuous, or to give an inadequate and ad-hoc explanation involving sequences of test functions.
Aside from the chapters on topological vector spaces and locally convex spaces, another glory of this book are the chapters on distributions, tempered distributions, and linear partial differential operators. The proof of Sobolev's lemma (Theorem 7.25) is meticulous. In the chapter on linear partial differential operators, the Malgrange-Ehrenpreis theorem and the elliptic regularity theorem are proved, and I think that this single chapter would teach one more about how to think about partial differential equations than Lawrence Evans' unwieldy monograph.
There is even a chapter on Tauberian theory, which gives probably the most structural proof of the prime number theorem that exists.
The rest of the book is on Banach algebras, in particular the Gelfand transform, and operators on Hilbert spaces. The spectral theorem is proved for unbounded normal operators in a Hilbert space, and results about strongly continuous one-parameter semigroups are proved, like the Hille-Yosida theorem.
The biggest requirement to use this book is first to know measure theory, both abstract measures and Borel measures. The next biggest requirement is the Cauchy integral formula from complex analysis. For both of these, it would be useful to have Rudin's "Real and Complex Analysis" on your desk while you read this book.
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