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Introduction to Analysis (Dover Books on Mathematics), by Maxwell Rosenlicht
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This well-written text provides excellent instruction in basic real analysis, giving a solid foundation for direct entry into advanced work in such fields as complex analysis, differential equations, integration theory, and general topology. The nominal prerequisite is a year of calculus, but actually nothing is assumed other than the axioms of the real number system. Because of its clarity, simplicity of exposition, and stress on easier examples, this material is accessible to a wide range of students, of both mathematics and other fields.
Chapter headings include notions from set theory, the real number system, metric spaces, continuous functions, differentiation, Riemann integration, interchange of limit operations, the method of successive approximations, partial differentiation, and multiple integrals.
Following some introductory material on very basic set theory and the deduction of the most important properties of the real number system from its axioms, Professor Rosenlicht gets to the heart of the book: a rigorous and carefully presented discussion of metric spaces and continuous functions, including such topics as open and closed sets, limits and continuity, and convergent sequence of points and of functions. Subsequent chapters cover smoothly and efficiently the relevant aspects of elementary calculus together with several somewhat more advanced subjects, such as multivariable calculus and existence theorems. The exercises include both easy problems and more difficult ones, interesting examples and counter examples, and a number of more advanced results.
Introduction to Analysis lends itself to a one- or two-quarter or one-semester course at the undergraduate level. It grew out of a course given at Berkeley since 1960. Refinement through extensive classroom use and the author’s pedagogical experience and expertise make it an unusually accessible introductory text.
- Sales Rank: #41637 in Books
- Published on: 1986
- Released on: 1985-02-01
- Original language: English
- Number of items: 1
- Dimensions: 8.50" h x .52" w x 5.32" l, .61 pounds
- Binding: Paperback
- 254 pages
- ISBN13: 9780486650388
- Condition: New
- Notes: BRAND NEW FROM PUBLISHER! BUY WITH CONFIDENCE, Over one million books sold! 98% Positive feedback. Compare our books, prices and service to the competition. 100% Satisfaction Guaranteed
Most helpful customer reviews
151 of 156 people found the following review helpful.
a bargain
By Dougabug
Buying this book is a no-brainer. It's only [price]. Unlike many dover books, it's not a reprint of a hundred year old text or dominated by 19th century style formal manipulations with indices running everywhere and no actual analysis or rigorous proofs. It covers the key topics of elementary real analysis (ie advanced calculus, or calculus with real proofs), which is basic to almost all fields of mathematics. In order to focus on the main ideas and definitions, it leaves out many topics and definitions (like functions of bounded variation, absolute continuity, lebesgue integration, differential forms, lebesgue covering lemma, f.i.c. etc.), so you should definitely have a more complete and comprehensive text like Apostol or Rudin (I prefer Apostol, although Rudin has complementary strengths). But for those omissions, Rosenlicht is quite a bit more readable than Apostol, which is best worked through methodically (since it contains a lot of results explicitly presented, its primary strength). Because it's both short and readable and focused, you can get through it reasonably quickly. Modern secondary school education teaches students to skip over the cursory material on elementary set theory, number systems, and strangely named but apparently obvious concepts like associativity and other field axioms, and to jump straight to the various sections with formulas in outlined boxes and procedure recipes for performing calculations. These rote recipes typically lack any conceptual unity and students are not often worse for wear forgetting or only superficially grasping previous ideas. This is a good way to never actually learn any legimate mathematics. Real math works on the principle of A->B->C->.... A proof (or your understanding of a mathematical concept) is only as strong as the weakest link in a chain. Learn and forget is not an adequate way to study math, and that's why introductory analysis is usually the course which weeds out those who will be able to explore the wider ocean of mathematical knowledge, and those for whom mathematics will be forever a giant forbidding question mark. The reality is, any above average person with the will, patience and time can learn this material and find themselves looking back thinking "Why was that so hard?" Blame it on the shabby state of secondary education. Do not skip over or gloss over the material on the real numbers if you aren't already fully conversant with the notion of l.u.b. and facile with its application (in which case why are you reading this book?). The l.u.b. property of the reals is your first real weapon to actually prove elementary theorem's of Calculus. Anyway this book isn't perfect, it's only good as an intro. and not a reference, but it serves the purpose of being accessible while still covering the key ideas and packing enough wallop to the requisite math whupping required by students numbed by years of inadequate secondary school education and the typical mechanical treatment of calculus and differential equations at the lower division level. And it's dirt cheap.
69 of 70 people found the following review helpful.
Solid Introduction to Metric Spaces and Continuous Functions
By Michael Wischmeyer
Introduction to Analysis by Maxwell Rosenlicht is another bargain from Dover Publications. I used this inexpensive mathematics reprint to help fill in gaps in my background before tackling more advanced mathematics. I found the first 150 pages to be challenging, but manageable. I had less success with the last 100 pages.
My college work was limited to applied mathematics, but in recent years I have developed some familiarity with metric spaces, topology, and analysis. (I previously reviewed Metric Spaces by Victor Bryant and Introduction to Topology by Bert Mendelson.)
Throughout his text Rosenlicht emphasizes how the same idea or theorem can be formulated in various ways. I found his approach to be quite helpful in clarifying more abstract representations of key ideas.
The first two chapters review set theory and the real number system and should be familiar to many readers. However, Chapter 3 (Metric Spaces) and 4 (Continuous Functions) are critical and require substantially more effort. My pace slowed dramatically.
For the reader new to metric spaces, Chapter 3 will likely be challenging, although metric space concepts are not really that difficult, just unfamiliar.
Rosenlicht demonstrates how statements concerning the open subsets of a metric space can be translated into statements concerning closed subsets, or alternatively into ones concerning sequences of points and their limits. Rosenlicht closes Chapter 3 with definitions and discussions of Cauchy sequences, completeness, compactness, and connectedness.
Rosenlicht begins Chapter 4 by illustrating that the familiar epsilon-delta definition of continuity of functions can be reformulated using the metric space open ball concept, or by using open subsets in metric spaces. He further explores the interdependence of theorems about continuity, limits, and convergent sequences. Chapter 4 concludes with discussions on continuous functions on a compact metric space and on continuous sequences of functions (analogous to sequences of points).
In chapters 5 (Differentiation) and 6 (Riemann Integration) we discuss the fundamental ideas of calculus using concepts and theorems introduced in the previous chapters. At this point I revisited a favorite calculus book by Salas, Hille, and Etgen. I was pleased to find that I now had greater insight into more advanced topics. Rosenlicht was indeed helping me.
Nonetheless, I had substantial difficulty with the longer and more complex proofs common in the remaining 100 pages, chapters titled Interchange of Limit Operations, the Method of Successive Approximations, Partial Differentiation, and Multiple Integrals. I again visited other textbooks, but this time looking for help with power series, the fixed point theorem, and the implicit function theorem. Although familiar with partial differentiation and multiple integration, I only skimmed the final chapters. I hope to return to Rosenlicht later after exploring another text on analysis.
I recommend Introduction to Analysis, especially for students looking for a review of analysis. This Dover reprint is a good buy, even if like me, you find the later chapters to be rather difficult.
26 of 28 people found the following review helpful.
An excellent introduction to real analysis
By callen@bc.cc.ca.us
A very readable book. Any student with two years of calculus should be able to read this with understanding. Starts with metric spaces, and proceeds through differentiation, integration, interchange of limits, implicit functions. A kinder and gentler introduction, yet rigorous.
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