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am 28. Januar 2000
I read with interest as one reviewer stridently urged students with no mathematical background to use Rudin as their introduction to higher mathematics. And I felt I must write something to stop some poor soul from being tortured by this book.
First, let me say right off that I like this book. I've seen better math textbooks though, so it doesn't get the five stars. The reason I like it is its utility as a reference. I can look up something to refresh my memory very easily. The style is concise, clearly outlined. Rudin also has some interesting proofs. I sometimes find myself looking up something pretty standard, and being enlightened at seeing familiar material in a new light.
But, and this is a big 'but', I wouldn't recommend this to a beginner. Mathematical 'maturity' is a funny thing. Some people have it; others don't. But most that do have it, get it by a long, arduous process of studying. Few are ready to immediately jump in and study the advanced textbooks. With this thought in mind, I think if you're reading this, wondering if this book is going to help you survive your first real math class, then Rudin will probably be tough. Not because it presumes some sort of secret knowledge. It doesn't. Just like any other intro analysis book, it doesn't assume you know analysis. But there are easier books. Like probably whatever your prof assigned for the class. 'Course, your prof could suck, and correspondingly, the book could suck. But if the faculty at your school is that inept, you're better off transferring.
Actually, I read with some surprise that some reviewers mentioned that the books they used had lots of useless pictures, etc. I don't recall ever reading an analysis book that had lots of pictures, period. So I guess I was lucky. But if your book is of that kind, then I guess I could recommend The Way of Analysis by Robert Strichartz. That's the book I first learned from. And it's the one used for the honors intro real analysis sequence(2 semesters) at my university here in Ithaca. The last half of the book is on applications of analysis to ODE's, Fourier series, Lebesgue integration. The style is very conversational, so it's definitely not a great reference book. I should add that the discussions are usually on motivating a definition, etc, rather than explaining something trivial, like a lot of similar books. Oh, and it has some pictures, but not many.
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am 22. September 2004
I am a fan of Rudin's books. This one "Principles of Matheamtical Analysis" has served as a standard textbook in the first serious undergraduate course in analysis at lots of universities in the US, and around the world.
The book is divided in the three main parts, foundations, convergence, and integration. But in addition, it contains a good amount of Fourier series, approximation theory, and a little harmonic analysis.
I loved it when I was a student, and since then I have taught from it many times. It has stood the test of time over almost three decades, and it is still my favorite. I have to admit that it is not the favorite of everyone I know.
What I like is that it is concise, and that the material is systematically built up in a way that is both effective and exciting.
The exercises and just at the right level. They can be assigned in class, or students can work on them alone. I think that is good, and the exercises serve well as little work-projects. This approach to the subject is probably is more pedagogical as well.
I think students will learn things that stay with them for life.
Review by Palle Jorgensen, Septembr 2002.
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am 9. Juni 2000
I feel puzzled to see, while the majority of reviewers are enthusiastic, some do not hide their "hostility" toward the author. Most of those who "oppose" complain that Rudin's style is not for beginners. In my opinion, this book is by no means "advanced", but WAS (back in my university years) suitable for students "contaminated" by tedious first-year calculus. I am sorry to say that if you feel Rudin's approach too abstract, then you will need more training to read any no-nonsense textbooks on mathematics.
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am 12. Dezember 1999
My first introduction to this book was in high school. I do not think that I have ever read a more fascinating book. Its clear explanations and exciting results engrossed me for hours at a time; I read and read and read; then I thought and thought and thought; and I repeated this for days at a time. Despite others complaints about Rudin's style, I found and still find this book wonderful. It may not be for everyone: it is, as one reviewer noted, in a Bourbakian style, and consequently does not always present much motivation. Any reader who finds this a problem should skip ahead a few pages to see where Rudin is going. Another aspect of this style is the very terse "outline" feel that another reviewer noted: Rudin has short proofs, no pictures, and fewer examples than many other analysis texts. This was not a problem for me; the proofs are very elegant and were an excellent introduction to the art of writing proofs. No difficult visualization is required; I was able to visualize all the pictures I needed in my head. The definitions are stated clearly enough that the student can construct his own examples, and the included examples are very good and clarify the most difficult portions of the text. The exercises, too, are very good. They range from trivial to mystifying, but all expand the reader's knowledge of the material. In summary, I think anyone who learns analysis should be exposed to this book. Reading it can be very difficult, and it is not unreasonable to spend hours on a page, but those hours will have been well spent.
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am 17. Juli 2000
I have taught Advanced Calculus and I regularly use analysis on the job as a scientist. I suggest that any new student or other interested professional consider either this text (Rudin), Border's Advanced Calculus (Dover, so a cheaper option), or Tom Apostle's Mathematical Analysis. As one of the top three, in my opinion, Rudin is outstanding - here's why: 1) the text is concise, but complete and readable for students that are properly prepared; 2) the style is traditional and provides a valuable introduction to the style used by professional mathematicians and scientists; 3) the range of topics is not overwhelming; 4) the problems are challenging, but, in my experience mostly solvable with a reasonable amount of effort; 5) it provides an excellent discussion on integration theory without attempting to do too much; 6) it does a decent job of introducing students to differential forms. Only "downside" is that the discussions on functions of several variables could use some elaboration and more "diagrams" to assist the student's intuition.
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am 1. Dezember 1999
I had a real analysis course from this text and I think it is a really excellent text which thoroughly succeeds at what it is attempting to do , namely to present the concepts of analysis in a crisp exact form and to achieve as much depth as possible in a limited space. It forces the reader to achieve a higher level of mathematical development to master its material. But I also understand how even an honest, hardworking student could be frustrated by the book and feel that it is not really helping them learn. I had an introductory calculus sequence from a text by Salas and Hille and an advanced calculus sequence from a text by Trench before I encountered Rudin in my real analysis sequence. I think it probably would have been a mistake to approach Rudin before I had been through these previous texts with more examples and illustrations. I think the text by Anton and the multivariable mathematics texts by Trotter and Williamson would have been even more beneficial for me to have studied before I attempted Rudin.
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am 9. März 1999
This book does not have a single picture or illustration in it. That is rare for a mathematical textbook (particularly analysis). You are forced, and rightly so, to form your own mental images of the mathematical objects defined and constructed within the text. This book is logical. Rudin lays it out in the definition, theorem, proof format, but does it in an amazing way. He leads you through a series of minor theorems and lemmas, you have no idea where he is going (unless you've already studied analysis) but then it all leads to a major result (for example the Heine-Borel Theorem in the Topology chapter) which is then used in the proof of many other theorems. Typical of how the book is written, he never tells you how important things like the Heine-Borel Theorem are, but the astute student soon figures it out. He does occassionally give a sentence or two of explanation or elucidation, but he mostly leaves that to the professor teaching the course. The exercises are excellent; tough and illuminating. Do as many as you can and you will learn a lot. If you can handle it, Rudin is the best way to learn Analysis (i.e., no BS). Good backround material before tackling Rudin would be Spivak's "Calculus" or Courant/John's "Introduction to Calculus and Analysis."
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am 24. Februar 1999
I first encountered Rudin's book as a graduate student in economics. At the time (ten years ago), it seemed to represent everything "wrong" with the economics curriculum (i.e. I hadn't gone into the field to ponder "compactness", "differentiability" etc.). However, I made a reasonable effort to read the book, but after 20 pages hadn't understood a single word (and my undergraduate mathematics background, although spotty in many places, was reasonably good - i.e. some exposure to projective geometry). Recently, partly because I had read Simon Singh's popular book on Fermat's Last Theorem (which I desperately bought in an airport before a transatlantic flight), I picked up Rudin's book again (exactly why, I don't remember). Much to my surprise, I suddenly seemed to understand the material that had tortured me! I particularly liked his treatment of the complex field, which had me fascinated for weeks (and still does). I also thought his treatment of the exponential, logarithmic and trigonometric functions was remarkable. I had never dreamed there could be such a remarkably simple structure underlying such mysterious numbers like "e", "i" and "pi". I don't claim to be a mathematician, but I am definitely hooked as an amateur. After reading Rudin (very non-linearly), I also picked up my old copy of Herstein's Abstract Algebra. Same reaction! I don't know what happened, but all the formalism suddenly makes sense to me. (Admittedly, these are books for the undergraduate, not professional-level texts.)
I do know what some of the other reviewers mean when they say Rudin's proofs are succinct, and don't (over)explain. But personally, I find the "gimmicks" and "tricks" fascinating, and feel they provide much food for thought. And if I don't understand some particular point, I don't sweat it. I'll never be a genius at doing proofs, but the tantalizing glimpses Rudin provides of the strange and wonderful world of mathematics are quite rewarding.
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am 15. Februar 2015
Walter Rudins Buch „Principles of Mathematical Analysis“ polarisiert seit Genartion seine Leser, die einen nennen es reine 'Bourbakisten Propaganda', andere möchten es kanonisieren; auf jeden Fall ist es eine der prägnantesten Einführung in die (reelle) Analysis in moderner Darstellung und wird oft als „Baby Rudin“ bezeichnet – in Abgrenzung zum „Big Rudin“ (Real and Complex Analysis).

Der Autor folgt in der Tat dem Prinzip der 'Denkökonomie' und führt die elementaren Konzepte der Analysis, wie Konvergenz und Stetigkeit, von vornherein abstrakt in Termen von metrischen Räumen ein. Die Darstellung ist präzise und streng, die Beweise sind ausgefeilt und ein Muster von Minimalismus, exakt, wenn auch Zwischenschritte gern dem Leser überlassen werden – insofern sind die 'Princples' auch ein wunderbares 'Arbeitsbuch' zur Analysis, der aufmerksame (und geduldige) Leser, wird die Auslassungen leicht füllen können.

Das Buch beginnt zunächst ganz traditionell, mit einer Einführung der reellen Zahlen als geordneter Körper mit kleinster oberen Schranke Eigenschaft, im Anhang zum 1. Kapitel wird die Existenz eines solchen Körpers mittels Dedekindscher Schnitte bewiesen. Es folgt ein knapper Abriss der Topologie, es werden metrischen Räumen eingeführt, und die Begriffen offener und abgeschlossener Mengen bereitgestellt, ferner werden die anspruchsvolleren Konzepte der Kompaktheit und des Zusammenhangs erläutert.

Die weiteren Themen sind: Konvergenz und Stetigkeit in metrischen Räumen und spezialisiert auf numerische Folgen bzw. Funktionen, Differentiation von Funktionen einer Variablen, Riemann Stieltjes Integrale, Funktionen Folgen und Reihen, sowie spezielle Funktionen – gemeint sind Exponential- , Logarithmus und trigonometrische Funktionen, Fourier Reihen und die Gammafunktion.

Der nächste Teil entwickelt die Theorie von Funktionen mehrerer Variablen: es werden Differentiale als lineare Approximationen und partielle Ableitungen betrachtet, das Inverse Funktionen Theorem wird mittels eines Fixpunkt Prinzips für kontraktive Abbildungen bewiesen, daraus werden das Implizite Funktionen und das Rank Theorem abgeleitet.

Die Entwicklung der Vektoranalysis auf der Grundlage von Differentialformen ist einem Analysis Einführungskurs nicht unbedingt üblich, Rudin wählt trotzdem diesen Weg – eines seiner 'Extras' --, das erlaubt es ihm, die Vektoranalysis auf ein wichtiges Theorem zurückzuführen; der Weg dahin wird nur minimalistisch ausgeführt: multidimensionale Integrale werden als iterierte Integrale einführt und einige elementar Eingenschaften bereitgestellt, danach werden k-Formen als Objekte auf n-dimensionalen Mengen, die sich auf k-dimensionalen 'Teilen' davon integrieren lassen 'identifiziert'; der Randoperator wird auf Ketten von k- Symplices betrachtet, damit sind sind Zutaten bereitgestellt , um den Satz von Stokes für k-Formen zu formulieren und zu beweisen. Daraus lassen sich – mittels einfacher Umformungen – alle üblichen Integralsätze der Vektoranalysis ableiten.

Note: die Abschnitte über Funktionen mehrerer Variabler und Differentialformen wurden in der vorliegenden 3. Auflage (1976) wesentlich überarbeite.

Das letzte Kapitel ist dem wahrscheinlich ungeliebtesten Kind eines Analysis Kurses – der Lebesgueschen Integraltheorie – gewidmet; ungeliebt wohl deswegen, da das Thema, gleich welche Weg mal wählt, langwierige Entwicklungen von Hilfskonstruktionen erfordert (Mengen Ringe und Algebren, mit entsprechenden Inhalten/ Maßen, deren Fortsetzungen, der Konstruktion geeigneter Funktionen, aus denen dann die integrierbaren herauszuschälen sind) , um am Ende eine recht magere Ausbeute an wichtigen Sätzen und Resultat zu erhalten – im Vergleich, was mit Riemannschen Integralen bereits möglich war. Dem ungeachtet, stellt sich der Autor der Herausforderung, allerdings wird das Material in diesem Kapitel – selbst für Rudins Maßstäbe – nur skizziert, etliche Eigenschaften werden als offensichtlich 'abgetan', d.h. Beweise sind Sache des Lesers. Auch hier gelingt es dem Autor, dass nach einem Parforceritt dem Leser schließlich die grundlegenden Eigenschaften von L^1 und L^2 Räumen zur Verfügung stehen.

Jedes Kapitel enthält zahlreiche Aufgaben, die zum Teil den stromlinienförmig formulierten Text um weiteres Material ergänzen und vertiefen.

Fazit: der „Baby Rudin“ ist ein exzellentes Lehrbuch zur Einführung in die (reele) Analysis, das in wohlüberlegt gedrängter Form etwa das Material eines einjährigen Kurses abdeckt, darüber hinaus noch etliche Extras enthält, die eine elementare Einleitung nach modernen Prinzipien abrunden. Es war und ist Grundlage oder Inspiration für zahllose Analysis Vorlesungen, und ist außerdem ausgezeichnet zum Selbststudium geeignet.
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am 4. Juni 2000
One gets the uncanny feeling that Rudin did not actually write the symbols and occasional words which grace his text, but rather recieved them, Moseslike, atop some mathematical Mt. Sinai. Imagine him descending from the heights, stone tablets in hand, irreversibly changed by his encounter with the SF, prepared to deliver the divine message to his people. But alas - at the foot of the mountain Rudin encounters a great congregation of heathen mathematicians. Poor Rudin watches in horror as these pagans unashamedly write intelligible proofs, motivate definitions, explain crucial concepts, sacrifice bulls, discuss mathematical history, etc. Shocked but still in possesion of his faculties, Rudin prepares to address these godless philistines. He raises the tablets above his head, and is on the verge of saying "Clearly,..." when out of the corner of his eyes, Rudin glimpses a wretched mathematician, dressed in filthy rags, commiting the most vile acts imaginable. At one point this base Yahoo even draws a diagram in the sand. The sight of such moral degeneracy inspires a wave a disgust in Rudin, who smashes the tablets to the ground and trudges back up the mountain. "Baby Rudin," as the Book is affectionately (sarcastically?) referred to, is a pedagogical nightmare, yet is inexplicably (and irresponsibly) used in most undergraduate analysis courses. This is especially sad when one considers that there are several excellent elementary analysis textbooks readily available. (most notably Strichartz' "The Way of Analysis.") Rather than helping students understand real analysis and where (and why) it fits into the mathematical landscape, Rudin seems to offer an absurd challenge to students - "I DEFY you to learn from my textbook!" Some mathematical masochists seem to enjoy this sort of thing (see many of the previous 35 reviews), but if you are one who enjoys the beauty and history of mathematics and can appreciate math not just as a tightly knit logical system, but as perhaps the highest flowering of the human mind - then you'll do well to keep far away from Rudin and his shameless textbooks.
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