Reviews (7)

The best undergraduate introduction to the subject
I can't think of a better place to begin learning functional analysis. The book is ideally suited for undergraduates or beginning graduates who have had one or two semesters of real analysis, linear algebra, and possibly topology. The author seemed extremely lucid with clear worked out examples. Phrases like "it's obvious" or "it should be clear" were not so frequent. It's quite a beautiful subject, with the last chapter providing a nice payoff application in terms of an introduction to quantum mechanics.

May be my only complaint was that the exercises seemed mostly five-finger ones. With that said, they should still challenge an undergraduate or beginning graduate, if not force them to re-visit the definitions and basic methods of proof.

I've always thought Rudin's "Mathematical Analysis" book deserved the title of "Best Undergraduate Math Text Ever", but this book has made me rethink that position.

The definition of classic
As shown in the first line on the cover:
Wiley CLASSICS Library

Reviews (73)

An excellent textbook
I think mathematics is a part of our culture.That's why, as a non-math major, I wandered into a very serious analysis class for mathematics majors.That might have been a disaster for me.Luckily, we used this book as a text, and it saved me.I read the whole book and diligently did all the exercises (of course, back then, it was the first edition, with only 227 pages and 140 exercises; it's somewhat more now).And that is my recommendation today.Read the book carefully and do as many exercises as you can.It certainly isn't easy.But it isn't, um, countably hard either.

The material in the book is self-contained.I guess that in theory, it could be mastered by any bright 14-year old who had learned some high school algebra and geometry.But I would surely recommend having much more mathematical sophistication than that as a prerequisite!

If you haven't learned the language of mathematics before, you'll enjoy the use of terms such as "countable," "real," "rational cuts," "measure," "ring," and "complete." By the end of the book, when the author claims that a proof (involving Cauchy sequences no less) is complete, you'll barely be able to suppress a desire to ask "Does every Cauchy sequence in the proof converge?"

In the first edition of this book, Rudin did mess up a little in his section on "the integral as a limit of sums." His theorem as stated was false.We cruelly dubbed it "Rudin's Last Theorem."Worse, he had used it "to prove some elementary properties of the Stieltjes integral."But that was all straightened out by the second edition.

I especially like the first couple of chapters.They give most readers the confidence to continue.And the final chapter, on Lebesgue integration, is very well written.One note of warning, though.Rudin begins this chapter by saying, "Some of the easier propositions are stated without proof.However the reader who has become familiar with the techniques used in the preceding chapters will certainly find no difficulty in supplying the missing steps."That is an exaggeration.It takes work.After all, this is, um, real mathematics you'll be doing!

I'm thankful that I was assigned this as my textbook.

A masterpiece
I absolutely agree with Professor Jorgensen.

I loved it when I was a student of physics, and I still love it because I tend to consider it as my personal standard in Classical Mathematical Analysis (and not only): sort of a "pacemaker" which sets the qualitative level to go back to just every time one is a little confused about what to do and where to go ;)

Reviews (21)

perfunctory treatment of stochastic processes
Tries to do too much and fails in delivering a rigorous understanding of the subject and this utter lack of mathematical rigor is the main flaw of the book.For a better understanding of stochastic processes consult the books by Karlin, Hoel and so forth.However, it does have some nice problems and some good examples to work through.I think that the author's other book, "A first course in probability" is much better.

Good but not great
This book tries to teach probability through examples.Most would be too difficult to work out as homework.This would be ideal for someone who wanted to get an understanding of Stochastic Process without investing too much time into homework.If, however, you wish to work out the homework, you will be frusterated to find out that there are not solutions to problems in the back of the book (except about 10% of the problems which are more or less worked out completely).If you are like me, you will want a book that gives answers to double check your work as you go along rather than the occasional complete solution.It is surprising that he did not put this in this book since "A First Course in Probability" included answers to the odd problems in the back.

Reviews (3)

fundamentals
Book is somewhat dated in the sense of application to current digital signal processing techniques, however it provides a good basic direction where the topic of information theory directs itself. At first appearance book seems overly complex, but is rather basic. Does not include advance topics such as pulsed modulation coding or topics such as constellation coding. A better source is to jump right in with Matlabs Communication toolbox.

A good book for learning both coding and information theory
It was a pleasure reading this book! I think Roman provides a freshintroduction to information theory and shows its inherent connections withcoding theory. I found his presentation on the noisy coding theorem verywell written. Another enjoyable part of the book is his treatment of linearcodes. He weaves a web on intrigue around it, showing it from differentperspectives (e.g. a design theory view, field , and vector space view).Twothumbs up!

Reviews (3)

great book
There are a lot of good introductory books on computation theory,
but after reading them you may be left asking "okay, what do
I read next?" Well _this_ is the book. If you're looking for something in between the undergraduate intro books and
the research-level articles then this is for you. It develops recursive function theory in a succinct, mathematically mature manner that isfreed from the details of any particular formalism. You should have previous exposure to turing machines and undecidable problems, an appreciation of the defense and use of Church's thesis, and familiarity with basic mathematical logic. Just to be clear, this book is NOT:
-a computer science, programming, or algorithms book
-an introductory book
-a book about automata or weak models of computation (such as regular languages or context-free grammars)
-a complexity theory book (no time bounds or np-completeness etc)

A classic!
The definitive book on computabilty and recursive function theory. Iremember reading this book in preparation forresearch in complexitytheory. I found it very stressful reading the book, but it was a good kindof stress. The kind that forces you to think deeply about what the authoris writing about. In addition to the main text, the author providesnumerous thought-provoking problems whose study could make a coure unto themselves. I read this book as a 3rd-year graduate student in

math. Ifyou are an undergraduate and are interested in computability theory, Irecommend Nigel's Cutland's book on thesubject.

Good reference
I looked long and hard for a reference in recursion theory and this wasthe only one which was acceptable.Luckily it is also quite good.

Mostbooks in the subject either introduce the material in their ownnon-standard notation which, while suitable for a survey course in thematerial is of little help when attempting to actually read papers in thefield.These books are also usually very basic ignoring things like thearithmetical hierarchy.Other books in this subject seem to mostly beadvanced texts and don't cover, or cover very briefly, the importanttheorems.

This book starts at turing machines and recursive functions. Going through the basic results like the halting problem and rapidly movingon to more advanced topics like creative sets, cylinders and hypersimplesets.Posts problem(with Friedberg's solution) and the fixed point theoremare covered as well.The final part of the book covers degrees ofunsolvability arithmetical hierarchy and the analytic hierarchy.

Whilethe book does cover recursive fucntions and turing machines I would suggestprevious experience with them before reading as the coverage is brief anddoesn't give the reader a feeling of how these systems work.

Reviews (9)

Deceptively Wonderful
OK here's the truth:This book is an awful text when accompanied by not so great prof is teaching from it (e.g. one who delivers nothing but the text). BUT... once you begin to understand enough to know that the "trivial" "exercise" and "left to the reader" proofs are quite straightforward, the book is probably the best reference in Algebra you can hope for.

Well worth the read for any budding mathematician
I've been acquainted with several introductory graduate algebra
books over the years, and prefer this one for its coverage of all the fundamental areas (groups, modules, rings, linear algebra, fields, and category theory), being concise, and providing great care when outlining each proof.

If one compare's the amount of material in this book to Jacobson's "Basic Algebra Vol 1", Grove's "Algebra", or Herstein's "Abstract Algebra", Hungerford's book gets the nod.
Moreover, I much more prefer the concise definition, example, theorem, proof format over the more colloquial approach, as can be found in Jacobson's text. For me at least, the payoff for reading an algebra text is the beauty found in the logic and reasoning from which very profound results arise from the complex interaction and use of more straightforward ones. And this is exactly where Hungerford's book shines through in tremendous glory. When outlining a proof he does an outstanding job in citating the results from previous Chapters that are used. For me this is the strength of algebra (In geometry I cringe when I get a picture for proof, and in analysis it is often quite complicated to verify that a given situation possesses the appropriate conditions needed to invoke some famous lemma or theorem).

One last good word about this book: I found the exercises both in abundance (after each section) and quite reasonable for a first year grad. student. Happy reading.

The bee's knees
This book is the Basic Language of Mathematics (by J. J. Schaffer) of the Algebra world.Without doubt it is an excellent dictionary of general facts about algebra.But learning by it will leave one with at best amusing memories and a nervous twitch.Just for a taste, "This proof has two parts.The first is easy.The second is left to the reader."About half the proofs in the book go like this.And so at the end of each section, the reader is left with just the dry theorems to attempt the exercises, without the slightest idea of how problems of a certain type are actually proven or even approached.And oh, the exercises.A few are easy.A few are open problems.The rest in between seem to at one point have been at the core of someone's respective masters thesis.

This book has three genuinely good uses.If you have a doctorate in pure Mathematics, a respectable doctorate that has nothing to do with PDEs and the thesis for which took longer to write on paper then it did to format the pictures to fit the margins, and you want to look up how much of the ring structure of R is inherited by R[x] in under 3 minutes, then this book belongs on your shelf.

If you have taken at least two algebra courses at the graduate level (Real graduate, not graduate equivalent.Most of my Algebra I class had two pretty good undergrad algebra classes coming in, and got slaughtered by Hungerford), then this book can make for a good review of basic algebra you should already know.

Editorial Review

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Reviews (2)

cryptic, brief, leaves you wanting much more...
Unfortunately, this book was used as the textbook in a mathematical statistics class taught by the author at UC Davis, which I was enrolled in.I found the book to be extremely brief, cryptic, and failed to even begin to explain the subject at hand in any kind of depth.I am an engineering student and am accustomed to technical material.This is the kind of textbook that gives the sciences a bad image.Being rigorous does not imply that you must be cryptic.Statistics is an exciting subject and this book utterly fails to inspire, explain, and enlighten.

Reviews (3)

More Rigorous Than Some Introductory Texts
I purchased Introduction to Metric and Topological Spaces two years ago. I was unprepared for its rigor.I am not a mathematics major, but I enjoy reading mathematics. My background includes calculus, linear algebra, differential equations, and other applied mathematics, but I have not had a course in real analysis. W. A. Sutherland intended this text as the next step after analysis.

After a brief foray, I retreated, placed Sutherland back on my bookshelf, and attacked some marginally easier introductory texts: Metric Spaces by Victor Bryant, Introduction to Topology by Bert Mendelson, and most recently, several chapters in Introduction to Analysis by Maxwell Rosenlicht. I periodically return to W. A. Sutherland's text to measure my understanding. I am now working on chapter five, Compact Spaces.

I doubt that Introduction to Metric and Topological Spaces would be foreboding to students that are familiar with real analysis. Sutherland understands that the abstractness and generalization can be difficult and shows concern with motivating the student. He repeatedly attempts to illustrate the value of generalization, especially in the study of continuity.

Sutherland often uses a lengthy series of examples of increasing difficulty to illustrate abstract concepts. In his discussion of metric spaces, we begin with Euclidian n-space metrics, and move on to discrete metric spaces, function spaces, and even Hilbert sequence spaces. He introduces open sets and topological spaces in a similar fashion.

The author occasionally suggests that the student might wish to make a geometrical diagram to help clarify some subtle point, but Sutherland includes few geometrical drawings in his text. His focus is clearly on proofs using the axioms of metric spaces and topological spaces.

Sutherland highlights sections that either require more knowledge of abstract algebra, or for other reasons are thought to be more severe.

Despite Sutherland's use of Introduction in the title, I suggest that any reader considering independent study might defer tackling Introduction to Metric and Topological Spaces until after completing a more basic text. Possibly a better title might be A Second Introduction to Metric and Topological Spaces.

An excellent, concise book on topology and metric spaces.
I enjoyed reading this book because of its clarity, conciseness, and niceway of relating topologicaland metric spaces. This book is ideal for thestudent who is learning about these subjects for the firsttime, whetheror not they intend to do more advanced work on the subject. The reader whowants to go on and learn about more advanced topics, should consultMunkres's book.

Reviews (3)

Good Introduction, too many typos
I am, what Prof. Bollobas would call a hobby mathematician. Some popular science book arouse my interest in graph theory, and the author of that popular science book recommended this book. I feel it was a vey good introduction to the subject, even though the proofs become challenging at times. His motivation for the subject is always concise but precise, one cannot but notice, that a master of the subject is writing about it.

The only distraction are the enormous number of typographical errors: I counted over 60, and this in a third corrected printing!?!

Excellent in content, but somewhat challenging in narrative
Bela Bollobas has the rare gift of having both deep mathematical insights, and the ability to eloquently communicate them in a way that is accessible to the average graduate student. In his book "Modern Graph Theory", Bollobas covers just about every exciting area of the subject, and does so in an up-to-date fashion that gives the reader a big picture of each sub-area of the field. The ability to do this not only seems difficult, but also essential, since he himself has written entire books on two of the chapters (extremal graph theory, and random graphs). Just about every major important theorem (including max-flow/min-cut Theorem, and theorems by Menger, Szemeredi, Kuratowski, Erdos/Stone, and Tutte) can be found here, and thus makes this book indispensable for anyone who does research in graph theory, combinatorics, and/or complexity theory.In my opinion the true highlights of this book are indeed those areas he knows best: extremal graph theory, random graphs, and random walks on graphs, the latter of which may be the best introduction to that subject that one will find in a textbook.

My only complaint, at the cost of perhaps half a star, is that his discussions and proofs often seem difficult to follow, as he will state something that to him seems quite obvious, yet to this reader often seemed a bit subtle, and would hence slow down the reading. Indeed, if these off-handed remarks were included as exercises at the end of each chapter, then the number of excercises would have swelled from the current 600 to well over one thousand ! Speaking of which, these 600+ exercises, although also representing another blessing of this book in that they add another degree of depth, tend to lack "starter" exercises, and go straight to the theory. But this is to be expected froma graduate text.

Finally, for the reader whose research significantly intersects with graph theory, but may not be ready or willing to be initiated by Bollabas into the world of graph theory, I would recommend Dietsel's graduate text on the subject. His book covers similar topics, but may be more clearly and transparently, but with less depth and insight.

A good introduction book
My profile is the following: I am a phD student in theoretical computer science and I needed a good introduction book to graph theory.

Reviews (9)

Restrained, Serious, Comprehensive: A Revolutionary Textbook
The best introduction to formal logic currently available, Ebbinghaus and Flum's *Mathematical Logic* drops the technique of "talking to cannibals" complained about by G.H. Hardy.Namely, Hardy couldn't do that anymore, and that was a long time ago: but for a long time readable logic texts (that is, texts not dedicated to the extrusion of logical categories from more copacetic algebraic ones) maintained a publicist's mien concerning the subject.Not so with this, but Ebbinghaus and Flum secretly maintain the publicist's attitude: this book covers more topics more thoroughly and elegantly than any of its competitors (including Wilfrid Hodges' much-touted, elsewhere-needed *A Shorter Model Theory*), including many topics of significance to graduate students in mathematics (infinitary logic, generalized quantifiers).

It really deserves to become the backbone of logic education -- that is, students deserve this and not half-hearted or whole-hog texts.For once, Springer is not blotting a superior product out of the market, but don't be confused: this is a rare occasion, as the serious aficionado of formal logic looks to North-Holland for guidance and Cambridge for avoidance.Not all "yellows" are equally mellow, and Springer's glut of nuvotexts is to be avoided like the intellectual plague it is -- many trends of the intellectual present cover up deeper perspectives with more appealing "localizations" not written all over them.

Reads like Mathematical Poetry
As others have pointed out, this book is not for beginners, but is very well suited for those with some confidence in formal logic and axiomatized set theory. The book is just great if you want to deepen your understanding of the subject beyond what can be had from undergrad level courses on the topic. It should be required reading for any student of computational logic.

The question this book addresses is not "why logic?", or "what is a formal logic?", but more specifically, "why is first-order predicate calculus with equality such a good foundation for mathematics?"

The formal mathematics is organized and presented so clearly and precisely that I felt I was admiring a fine crystal structure.
The notation used may seem excessive to some, but it actually is the least amount of notation that could be gotten away with without resorting to glossing over fine distinctions.For example, many logic books assume a fixed countably infinite number of function and predicate symbols, which leads to some confusion when comparing different axiomatizations of the natural numbers, or of groups.This book on the other hand is crystal clear on how such different axiomatizations are related to each other.Another subtle point I never noticed before about first-order predicate logic but that is pointed out in the footnote on page 73 is that one might think it possible that just because a formula can be proven with one choice of predicate and function symbols, it might not be provable with a different choice of symbols.It turns out that this cannot happen as a simple consequence of the completeness theorem! (p. 85)

The book explores second-order predicate logic and makes explicit some of the difficulties, such as incompleteness and even the problem of how closely the truth of a formula in second order logic depends on what we take as true in set theory: different axiomatizations of set theory lead to different semantics for second-order predicate logic!

There is a great chapter on the incompleteness theorems, and in addition to Goedel's theorems, there is a section on Register Machines (a version of Turing Machines) and a proof of the undecidability of arithmetic using the halting problem, as well as a more general theorem about the undecidability of any theory that can encode the workings of a Register Machine.

The next section is a reasonable presentation of the mathematical underpinnings of logic programming.

The book concludes with an algebraic characterization of elementary equivalence followed by two deep theorems by Lindstrom that demonstrate the uniqueness of first order predicate calculus among formal languages with set theoretic semantics.