A Problem Book in Real Analysis (Problem Books in Mathematics) 2010th Edition by Asuman G. Aksoy (PDF)

Description

Education is an admirable thing, but it is well to remember from time to time that nothing worth knowing can be taught. Oscar Wilde, “The Critic as Artist,” 1890. Analysis is a profound subject; it is neither easy to understand nor summarize. However, Real Analysis can be discovered by solving problems. This book aims to give independent students the opportunity to discover Real Analysis by themselves through problem solving. ThedepthandcomplexityofthetheoryofAnalysiscanbeappreciatedbytakingaglimpseatits developmental history. Although Analysis was conceived in the 17th century during the Scienti?c Revolution, it has taken nearly two hundred years to establish its theoretical basis. Kepler, Galileo, Descartes, Fermat, Newton and Leibniz were among those who contributed to its genesis. Deep conceptual changes in Analysis were brought about in the 19th century by Cauchy and Weierstrass. Furthermore, modern concepts such as open and closed sets were introduced in the 1900s. Today nearly every undergraduate mathematics program requires at least one semester of Real Analysis. Often, students consider this course to be the most challenging or even intimidating of all their mathematics major requirements. The primary goal of this book is to alleviate those concerns by systematically solving the problems related to the core concepts of most analysis courses. In doing so, we hope that learning analysis becomes less taxing and thereby more satisfying.

User’s Reviews

Editorial Reviews: Review From the reviews:“Spread out over 11 chapters, this is a collection of 319 problems in what used to be called Advanced Calculus. … The authors see their book primarily as an aid to undergraduates … but I view it as being helpful to teachers in supplementing their courses or in preparing exams. … However, kept on a course reserve shelf of an academic library, the book under review might entice and benefit the more dedicated student. It certainly merits the attention of instructors of elementary analysis.” (Henry Ricardo, The Mathematical Association of America, June, 2010)“A very readable collection of interesting problems of varying levels of difficulty. It is intended to build a bridge between ordinary high school or undergraduate exercises and more difficult and abstract concepts or problems. The book is so delightfully written that anyone who simply likes working on challenging problems could read it independently. … recommends this book to all students curious about elementary real analysis and how to learn it through solving problems. … a welcome resource for organizing their activities at a good level.” (Vicenţiu D. Rădulescu, Zentralblatt MATH, Vol. 1186, 2010) From the Back Cover Today, nearly every undergraduate mathematics program requires at least one semester of real analysis. Often, students consider this course to be the most challenging or even intimidating of all their mathematics major requirements. The primary goal of A Problem Book in Real Analysis is to alleviate those concerns by systematically solving the problems related to the core concepts of most analysis courses. In doing so, the authors hope that learning analysis becomes less taxing and more satisfying.The wide variety of exercises presented in this book range from the computational to the more conceptual and varies in difficulty. They cover the following subjects: set theory; real numbers; sequences; limits of the functions; continuity; differentiability; integration; series; metric spaces; sequences; and series of functions and fundamentals of topology. Furthermore, the authors define the concepts and cite the theorems used at the beginning of each chapter. A Problem Book in Real Analysis is not simply a collection of problems; it will stimulate its readers to independent thinking in discovering analysis.Prerequisites for the reader are a robust understanding of calculus and linear algebra.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Background: I am a graduate student at Stanford, brushing up on Analysis. I purchased this text in hopes of having well-written problems and corresponding solutions.Unfortunately, the solutions are often times unnecessarily complicated. Further, the notation is inconsistent in the problems and solutions, and there are quite a few typos that get in the way.Many of the exercises are identical to what you will find in other Analysis books.I wouldn’t recommend this book to somebody looking for more practice problems. There are much better resources out there. Consider “Understanding Analysis”, by Abbott, or “Elementary Analysis” by Ross if you’re looking for an easy introduction to real analysis in one variable.

⭐Only the looking and typing is really nice, but there are so many books those are better than this one in term of contexts. Simply speaking, I believe, finding books of similar topic in bookstores can always get some which are much better than this one.

⭐The worked solutions in this book taught me how to write a proof. Starts with sets and induction and moves well into Integration and even covers a fair amount of Topology. Covers all the necessary definitions and from them you must prove most important theorems. Because of this feature, I believe this book could also be used as standalone text for those with initiative.

⭐A MUST for taking a real analysis (intro-level) class.One reason that student suffer in real analysis is thatthere is neither step-by-step instructions, nor enough examples.This books serves as an excellent source of examples.Highly recommend this !

⭐The book presents 11 Chapters in the subjects: Elementary Logic and Set Theory, Real Numbers, Sequences, Limits of functions, Continuity, Differentiability, Integration, Series, Metric Spaces, Fundamentals of Topology, Sequences and Series of functions. At the beginning of every chapter the author has presented in brief the basic definitions of the concepts used in the statement and solutions of the problems which follow the respective chapter. Most of the problems of the book are well-known in other books devoted in Calculus and Real Analysis. The solutions provided are clear and useful for a better understanding of the theory. I believe that this book can be of help mainly to undergraduate students who wish to learn further material that is not included in a Calculus book.

⭐I am taking an analysis this quarter, and this book is helping me survive the course. Analysis is all about writing proofs, and there is really no other way to learn how to write proofs other than to try problems and compare your answers to the solutions. This book has many problems, and the solutions are broken down step by step so that they are really easy to follow. An extra bonus was that certain problems I did in this book showed up on homework assignments and the midterm. It seems the authors have a good sense of the types of problems featured in a standard analysis course, and the types of problem’s analysis teachers like to ask on exams and homework. Especially if you find your primary textbook hard to follow (like I did), I highly recommend this book.

⭐My math teacher recommended I buy this book for extra practice. There are a ton of problems, and the solutions are usually easy to follow. Also, although this is not that important, the book is really well bound, has a hardcover, and seems more like a textbook than a problem book. A little pricey, but well worth it..

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