Resources For The Study of Real Analysis available in Hardcover
- Pub. Date:
- American Mathematical Society
This book is a collection of materials gathered by the author while teaching real analysis over a period of years. It is intended for use as a supplement to a traditional analysis textbook, or to provide material for seminars or independent study in analysis and its historical development. The book includes historical and biographical information, a wide range of problem types, selected readings on a variety of topics, and many references for additional study. Since all these materials are collected into a single book, teachers and students can easily choose items most suitable for their purpose. Teachers may use the book as a supplement to their courses, while students may read much of the book on their own. No other book has been written specifically as a supplement for a real analysis course.
Table of Contents1. Review of calculus; 2. Analysis problems; 3. Essays; 4. Selected readings; Annotated bibliography; Additional references; Index.
What People are Saying About This
"Brabenec has provided a rich smorgasbord or mathematical analysis-including a host or problems, historical essays, and selected readings-from which no one should go away hungry."
"The book offers even more than its title suggests: a true trove of resources for students (and their teachers) who face the exciting-but often rough-passage from the routing calculations of elementary calculus to the deeper arguments and insights of real analysis. There are gems here for average students who need review; for the quickest students who need mathematical challenges; for teachers who need ideas; and for everyone with a taste for mathematical highlights and culture."
St. Olaf College
This book is a collection of materials I have gathered while teaching a real analysis course every year for more than thirty five years. I prepared it with the hope it will benefit and enrich the experience both of students, who take a real analysis course, as well as those who teach it. This collection is intended to supplement a traditional real analysis textbook, where teachers and students may choose items of interest to them. It is my conviction that such supplementary materials have a much greater chance of being used in an analysis course if they are readily available in one place.
Part I contains materials that provide the greatest benefit if read before the real analysis course begins. Because such a course assumes knowledge of topics from calculus of one real variable, review of this material in advance frees the student to concentrate on new content and theoretical emphasis in analysis without having to revisit the calculus at the same time. The outline of a traditional calculus course allows a student to check necessary topics for review, whereas the calculus review problems offer the opportunity to refresh necessary skills which may have lain dormant for some time. Many hints are provided with the problems to encourage students to carry out this review.
The problems in Part II are intended to supplement the ones usually found in an analysis text. There is a wide variety of problem types, and each one has exercises for the student to attempt. Most of them contain explanatory detail about the historical background of the topic or how it fits with other parts of analysis. I try to present a topic from a variety of perspectives in order to enhance learning and understanding.
The entries in Part III are called essays, and represent short discussions of a particular topic from calculus or analysis. Some of these are content-oriented, whereas others are intended to give an alternate perspective after the student has first studied the material in a traditional manner. The collection of six essays based on time periods presents a historical overview of the development of analysis by concentrating on the main individuals who were influential in this development.
Part IV contains a collection of five supplementary readings to illustrate the variety of materials that are available beyond a standard textbook. The annotated bibliography contains many references that can be read with profit by faculty and students. There is information for each one to help readers decide which might best fit their interests and needs. We are fortunate to be living at a time when there is a growing interest in these kinds of supplementary materials 0 the past twenty years especially have seen many new entries of a historical, biographical, or expository nature.
Let me explain how I use these materials in my analysis course. I encourage students to work on the calculus review problems before the analysis class begins. Since our course is taught in the fall semester. I give them the problems before they leave for summer vacation. At the beginning of the course, while presenting the abstract material on properties of real numbers and sequences, I use Essays 7 and 8 to review the material on derivatives and series from calculus. This not only gives students a more familiar alternative to the abstract material, but it also provides an introduction to the spirit of careful organization and attention to detail that is essential for work in analysis. Students read one of the six historical essays a week and we spend some time in class discussion on the materials. I use Essays 9 and 10 in the middle of the course to provide an alternative perspective for he topics of proofs and topology, and assign supplementary problems from Part II at appropriate places in the course. The material in chart form from Problems 8, 9, and 20 works well in class discussions. I like to assign some of the enrichment problems in Section 3 to teams of students and have them learn the materials, write up solutions to the exercises, and make oral presentations to the rest of the class. Students enjoy this opportunity to make presentations.