- Series: Higher Mathematics Series
- Hardcover: 483 pages
- Publisher: McGraw-Hill Education; 3 edition (May 1, 1986)
- Language: English
- ISBN-10: 0070542341
- ISBN-13: 978-0070542341
- Product Dimensions: 6.8 x 0.8 x 9.5 inches
- Shipping Weight: 1.7 pounds (View shipping rates and policies)
- Average Customer Review: 40 customer reviews
- Amazon Best Sellers Rank: #548,140 in Books (See Top 100 in Books)
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Real and Complex Analysis (Higher Mathematics Series) 3rd Edition
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Also, Rudin does not discuss some of the more advanced or interdisciplinary topics such as distribution theory (Sobolev spaces, weak derivatives, etc.) or applications of measure theory to the probability theory, both explored in the book by Folland. Last but not least, it's worth noting that contrary to the common practice, Folland includes many end-of-chapter notes where he outlines some important historical aspects of the development of the topics, and also gives a few references for further study. For example, in the notes section at the end of the chapter on Lebesgue integration, he mentions --and briefly outlines-- the basics of the theory of "gauge integration" (aka Henstock-Kurzweil theory) which serves to construct a more powerful integral than that of the Lebesgue's. As another instance, having already defined and used "nets" within the chapter on topology, in the end-notes Folland also introduces "filters" and "ultrafilters". These are all machineries which have been developed to play the role of the metric space sequences in general (locally Hausdorff) topological spaces, but for some historical reasons, ultrafilters have nowadays taken a backseat to the nets (the older general topology books usually prove the Tychonoff theorem using ultrafilters). All said, I can recommend taking up Royden as your very first approach to measure theory, then based on how well you think you have learned the first course, move on to either Rudin or Folland for a more advanced treatment. Please note that the other books I have mentioned above do not discuss complex analysis, a subject which is also masterfully presented in Rudin. There are however a few other equally well-written complex analysis books to pick from, for example John B. Conway's classic from the Springer-Verlag graduate text series, or L.V. Ahlfors's wonderful monograph, to name just a couple.
The simplicity of the organization and the efficiency of the proofs really cut down on the clutter, noise and distractions in focusing on major aspects to consider in this basic area. However, I find that Rudin, beyond the simple, sparse organization and the elegant proofs, really does not display much interest in helping the student struggle through the profundity of this beautiful field. One moves very quickly, in the book, from the concrete, intuitive ideas of Riemann and Lebesgue, to very deep insights derived from the brilliant work of Urysohn, and the creative insights of Riesz. One cannot over-estimate the struggle involved in grappling with the ideas in Rudin: They are deep, deep, deep. However, for most of us, that is the glory of mathematics, and over the years, my insights into this field have grown, due in part to this wonderful book. Working in isolation as an amateur mathematician has, no doubt made the climb to the current work difficult. The mathematical field seems to have simply exploded in complexity and breadth since 1940 or so. Nevertheless, many of the basic insights were arrived at in the first half of the twentieth century. Therefore, I think that a book like Rudin's remains valuable, despite the enormous strides in progress mathematics achieved in the latter part of the twentieth century. I truly feel that progress in math cannot be guaranteed in this distracting world. An opportunity to focus on math, really, can come very rarely in life, especially for an amateur. This makes sparse books, that are highly focused, and not directed at a trivial level very important, even if it describes only classic work of a distant past. I will say, however, with all the many roads one can travel from Rudin's work, it is wise to have some goal to work toward in modern mathematics. Perhaps I have simply been humbled due to the many struggles I have faced, but I find the panorama of modern mathematics very overwhelming.
Chapter one develops the abstract integration theory, by establishing the Lebesgue monotone convergence theorem, and the dominated convergence theorem. The extension this is making to Riemann's approach is not entirely clear, at this point, as we assume that we are simply given a measure, and, in this first chapter, we are not really trying to understand how measures can arise. However, one does become aware of the Lebesgue approach, in partitioning the range space, rather than the domain space, as in Riemann's approach. The Lebesgue approach is recursive: Far more universal in character than Riemann's. Also, Rudin makes it fairly clear that this is a "topological" approach, whereas the Riemann method is not utilizing this more abstract framework. One consequence of this is that sets of measure zero are more important than for Riemann's theory, and Rudin makes it clear that this entails quite a shift in thinking from our ordinary intuitive notion of the length of an interval. One recognizes that the Lebesgue theory seems much more capable of dealing with sequences of functions than the Riemann theory.
Once one realizes that the topological approach and measures are extremely important, with respect to Lebesgue's theory of integration, the next step is to try to demonstrate how measures can evolve out of topological frameworks. The Lebesgue approach alone, with its recursive perspective, certainly sets the stage for this development. However, utilizing the vector space structure of the range space, and a close analogy with continuous functions requires a further development, represented by the Urysohn lemma, in that this lemma permits one to "pullback" the structure of the range space, and complete a picture of how measures can arise within a topological framework. This is the goal of the second chapter of Rudin's book, and one is treated to a royal feast of a beautiful formulation of the Riesz representation theorem. This is, without a doubt the highlight of Rudin's entire development of real analysis. Concluding this chapter is a discussion of Lebesgue theory in Euclidean space, showing that the Lebesgue measure inherits some of the geometric features of Riemann's theory, and opens enormous vistas through the achievement of the Riesz representation theorem. Two important theorems showing the close connection between measurable functions and continuous functions conclude the chapter. This chapter alone is a course in real analysis and wonderful, worthy of considerable time to digest and appreciate. A number of the problems, in this regard, at the end of the chapter, are important in terms of exploring the meaning of the Riesz representation theorem. However, on the whole there are many fewer problems than one really needs for students to develop an appreciation of what this chapter represents and the large step in progress achieved through adding the Urysohn lemma into the mix with Lebesgue's approach. The first two chapters alone are worth recommending Rudin's text as a "five star" book.
The third chapter introduces Lp-spaces, including L-infinity. The development is straightforward, but Rudin does benefit somewhat in having introduced the Riesz representation theorem. The Lp spaces are shown to be metric spaces (via the Holder inequality used to prove the Minkowski inequality), and then an application of Fatou's lemma shows these spaces to be complete. Simple functions and continuous functions with compact support are shown to form dense subsets in these spaces. This is an application of the Lebesgue dominated convergence theorem, and Lusin's theorem. The closure of the continuous functions with compact support, with the supremum norm, is shown to be the continuous functions vanishing at infinity, by an application of Urysohn's lemma. Since this is a standard development, its simplicity and elegance can easily go unappreciated. It is worth comparing Rudin's development here with some others. Although very standard, I still prefer Rudin's organization.
Rudin starts to discuss Hilbert space theory in Chapter 4. His discussion culminates with the Riesz-Fischer theorem. His proof to this theorem is a model of efficiency, based on a little lemma that is structured so nicely it practically proves itself. He also wants to apply the abstract results to the case of trigonometric polynomials. This requires showing that these polynomials are dense in the set of periodic continuous functions. For this, he uses a proof very similar to a proof of Fejer's theorem, which is a result associated with arithmetic means of partial sums of Fourier series. One can develop some understanding and appreciation for the proof, which otherwise would seem more like a magic trick, by recognizing that a central element of the proof is the construction of an "approximate" delta-function. I wish he had discussed Fejer's theorem more, because the point of view from that theorem is so critical to appreciating the proof. Of course, in PMA, Rudin does discuss Fejer's theorem quite a bit more thoroughly.
Banach space theory is the topic of Chapter 5, where three of the basic results from that theory are proved: The uniform boundedness theorem, the open mapping theorem and the Hahn - Banach theorem. I found that it was helpful to phrase the theorems in my own words, and to supply proofs in the same style as well. In addition, I consulted some outside sources. The proof to the Hahn-Banach theorem is very simple, and therefore deceiving, in that there are a few hidden subtleties. Rudin applies the uniform boundedness theorem (or Banach-Steinhaus theorem) to the problem of pointwise convergence of Fourier series, with quite satisfying results. He considers an analogue to the Riesz-Fischer theorem in L1, which applies the open mapping theorem. In the case of the Hahn-Banach theorem, he develops a nice abstract integral representation theory, which he brings down to a concrete approach to the Poisson integral. This chapter beautifully illustrates his approach to Lebesgue theory, which is one of a very general viewpoint and shows how Lebesgue theory is not just of interest for Lebesgue integration, but allows one to develop a useful framework for functional analysis. He displays considerable insight in this chapter that repays close study.
Rudin completes, in Chapter 6, the discussion of measure theory, started in Chapter 1, and culminating in Chapter 2 with a version of the Riesz representation theorem. He starts with a beautiful proof of the Radon-Nikodym theorem, inspired from the work of von Neumann, and centered around the association of measure with vectors in Hilbert space. The brilliance of this may be hard to appreciate if one is encountering real analysis for the first time. The remainder of the chapter is dedicated to consequences of the Radon-Nikodym theorem. The two final results are quite significant and discuss how complex measures can arise from linear functionals. Taking the perspective associated with von Neumann's proof of the Radon-Nikodym theorem, these are not so surprising as they might otherwise seem. The last result is another version of the Riesz representation theorem that applies to complex measures. Reading this chapter was like reading a whole book, and it deserved more time than I was able to dedicate to it. Both the ideas and the technical details in this chapter seem important.
The seventh chapter concerns differentiation. Much of the development is pretty technical and relies on the Radon-Nikodym theorem. The first part of the chapter is quite long, with Rudin struggling with a number of technical subtleties, and concerns the differentiation of measures (with respect to Lebesgue measure). Rudin achieves, in this regard, the satisfactory and plausible type of theorem that one might expect. In the next part of the chapter, Rudin comes at the fundamental theorem of calculus in a couple of different ways, one associated with the differentiability of absolutely continuous functions. The last section of the chapter concerns the change of variable formula, using the Jacobian. Due to the technical considerations, the material in the chapter, while resulting in conclusions that are satisfactory, can be difficult to follow. I suggest avoiding the technicalities and focusing on simplification, main points and on approximations, instead of exact inequalities, as Rudin does. This can be as rigorous, or nearly as rigorous as Rudin's development, without the technical clutter impeding one's understanding. A focus on a technicality can often result in much effort exerted at achieving a rather minor result.
Rudin's treatment of integration on product spaces is given in Chapter 8, where he discusses the Fubini theorem using an approach that is not very illuminating, despite being technically efficient. Following this discussion, he gives two applications of the theorem. In one, he establishes the existence of the convolution integral in a certain case where the motivation is pretty clear, from considering L1-norms. The second application presents a theorem due to Hardy and Littlewood from 1930 that is quite technical involving maximal functions, but the proof relies on an interesting technique due to Marcinkiewicz that is worth studying.
Chapter 9, after a few preliminaries, gives a very pretty application of the preceding theory to Fourier transforms. The main result is a version of the Plancherel theorem, concerning Fourier transform as an L2 isomorphism. However, after this, Rudin also develops a characterization of the translation-invariant subspaces of L2, and in the final section of the chapter presents a theorem concerning homomorphisms on L1 that relates them to Fourier transforms. For a relatively short chapter, it is just packed with interesting ideas and techniques. It is a beautiful culmination of his work on real variables.
Rudin presents many elementary results in complex analysis in Chapter 10, as he starts discussing that subject. Nevertheless, although somewhat more conventional than some of the work in the preceding chapters, and also, somewhat peripheral to his prior main focus in developing Lebesgue theory, the chapter is still worth study. He retains the elegance of style and the simple, direct proofs that makes his work appealing. There are many "name" theorems in this chapter: Cauchy, Morera, Liouville, Maximum Modulus, Parseval's Formula (applied to this special case), Fundamental Theorem of Algebra, Cauchy Estimates, Open Mapping, Residue Theorem and Rouche's Theorem. He also presents a computation of the Fourier transform of sin(x)/x that is somewhat interesting, and can be compared with less formal work elsewhere. The chapter is basically a basis for more technical work to follow.
I definitely feel that Chapter 11 on harmonic functions has been one of the most important chapters in this early development in Rudin's book. There are many aspects one could point to in this chapter as outstanding, but one sees, in particular the development of a very nice theory elaborated by Hardy and Littlewoood in a 1930 paper. I know this is "ancient", and the other topics discussed in the chapter were all satisfactorily elaborated prior to 1940. But I think this should not lead one to slight the importance of this seminal period from about 1900 to 1940 when functional analysis was taking shape. In addition, it is easier to relate this "pure" math to physics than the later work done in functional analysis. I think there is a tendency to overlook Hardy's work due to his remarks suggesting the irrelevance of pure mathematics, but I find his work to be both deep and fine, and worth studying if one has interests in physics and certain other applied areas. Perhaps this is only an indicator that what was pure mathematics in 1930 has become important in applications today.
Chapter 12 offers an extended development for the maximum modulus theorem. At the end of the chapter, a partial converse to the theorem is presented, which is quite pretty. Based on this, Rudin gives a brilliant proof (due to R. Kaufman), which is simply stunning, and a real gem. There are other very interesting discussions in this chapter as well. There is, for example, a very nice application of convexity in Rudin's discussion of the Phragmen-Lindelof method.
More discussions of the mathematics between statements of theorems would have been appreciated, to help the student understand the theoretical framework, how a particular theorem fits into the theory, and why it is important, as well as help the student develop a perspective and encourage the student (if that student has time) to read other references and recent original papers related to the material. I have found Rudin's Notes and Comments toward the end of the book to be somewhat instructive. He gives there various useful references, for example to Carleson's theorem.
It could be argued that Rudin presents too many proofs, and sacrifices a great deal of clarity (that I think would be appreciated by most students) for the sake of efficiency. There are pros and cons to this. In providing well-crafted, efficiently organized proofs, he makes his book a useful (if sometimes sterile) model for students, and worth retaining as a reference. There are some beautiful and deep ideas in the proofs he gives that are worth pondering (if one has time). On the other hand, caving in to clarity would have greatly expanded the didactic value of his book, and with respect to proofs, often just a hint would more than suffice for a student to grasp what is involved. I think that a student is perfectly capable of making good progress or even proving (perhaps not as well as the models Rudin's proofs provide) many of the theorems Rudin states. After presenting a theorem, a discussion, without going into detail, of his own proof or a proof he prefers would be worthwhile.
Although Rudin's proofs can be very illuminating to read, a lot of times I find it is as if Rudin is "pulling things out of his hat", and there is a whole context that he brings in with his proofs that do not always relate directly to what he is developing theoretically. Furthermore, I find it is hard to appreciate his "tricks" unless one has struggled and proved a theorem oneself.
Rudin relegates to the problems, sometimes, results that fit in quite nicely with the theory his is developing, and would better have been stated in the main part of the text, but perhaps without proof. I certainly felt, for example, that Egoroff's theorem was sufficiently important to his main line of development that it would have been worthwhile to include that as part of a chapter rather than relegating it to a problem.
The above criticisms are in no way meant to detract from the brilliance of Rudin's work. His book is a masterpiece, but this does not mean it would not be wise to consult other references. As a graduate student, being introduced to analysis through this book, I found it daunting and not helpful. Now that I am older and can appreciate Rudin's viewpoint, I find the theoretical framework supplied by this book to be beautiful, and also find the book to serve as a very good reference for the basics.