Yet Another Introduction to Analysis

By Victor Bryant

Book Overview

Mathematics education in schools has seen a revolution in recent years. Students everywhere expect the subject to be well-motivated, relevant and practical. When such students reach higher education, the traditional development of analysis, often divorced from the calculus they learned at school, seems highly inappropriate. Shouldn't every step in a first course in analysis arise naturally from the student's experience of functions and calculus in school? And shouldn't such a course take every opportunity to endorse and extend the student's basic knowledge of functions? In Yet Another Introduction to Analysis, the author steers a simple and well-motivated path through the central ideas of real analysis. Each concept is introduced only after its need has become clear and after it has already been used informally. Wherever appropriate, new ideas are related to common topics in math curricula and are used to extend the reader's understanding of those topics. In this book the readers are led carefully through every step in such a way that they will soon be predicting the next step for themselves. In this way students will not only understand analysis, but also enjoy it.

• Format: Paperback
• Language: English
• ISBN: 052138835X
• ISBN13: 9780521388351
• Release Date: September 1990
• Publisher: Cambridge University Press
• Length: 300 Pages
• Weight: 1.03 lbs.
• Dimensions: 0.8" x 6.1" x 9.1"

Customer Reviews

Yet Another Good Text from Victor Bryant - Great for Self-Tutorial Purposes

Victor Bryant's informal, conversational text, Yet Another Introduction to Analysis, offers an engaging, well-motivated introduction to real analysis, but it is not a full substitute for a more formal, more axiomatically structured approach. However, Bryant's text is a great companion text, and is especially suitable for self-tutoring purposes, or as pre-read prior to taking that first rigorous analysis class. The reader need only be familiar with first year calculus. As is so often said, mathematics is not a spectator sport, and Bryant clearly expects his readers to work the problem sets; the text frequently makes direct use of the results of previous problems. Bryant provides full solutions to nearly every problem, another reason why this book is so good for self-study. (The solutions section is 67 pages.) Bryant's problems were rarely difficult or overly time consuming, and are most notable for clarifying key points in the text. Bryant begins with a brief examination of real numbers, looking at why the irrational numbers so out number the rational ones. (The completeness axiom is introduced in the short first chapter.) I particularly enjoyed the next section, Bryant's examination of whether a series converges or not and ways to determine the sum of an infinite series. (I had not previously been all that interested in the study of series, but Bryant's approach peaked my interest. I have now purchased a more advanced Dover reprint, Infinite Series by James M. Hyslop, for follow-up reading.) A longer section examines the familiar concept of a function from various perspectives, using the inverse relationship between exp and the log as one of the key examples. The final two chapters focus on a primary topic of analysis, the basic theorems of differentiation and integration. Familiarity with partial differentiation and multiple integration is not needed. Some readers may find Bryant's conversational approach to be too wordy and occasionally digressive, but I personally enjoy his leisurely style. I also recommend Bryant's short text titled Metric Spaces, Iteration and Application, published by Cambridge University Press. Another good choice is Maxwell Rosenlicht's Introduction to Analysis, available in an inexpensive Dover edition. It offers a more traditional, structured approach to analysis that is suitable either as follow-up to Yet Another Introduction to Analysis, or as a stand-alone self-tutorial text. Although Rosenlicht's text emphasizes generality and abstraction to a greater extent, it is still more concrete and less terse than many standard texts on real analysis.

Accessible book gets to the heart of analysis

Bryant's book on analysis is a great illustration of what a textbook should be. He takes what many upper level college mathematics students consider to be the most tedious and boring topic - analysis- and presents it in a clear, interesting and effective way. Calculus at the college undergraduate level is usually taught in 3 semester long classes where integrals and derivatives are seen as tools for finding areas under curves, or volumes of objects, etc. which is the way engineers are made to view Calculus during their scholastic careers. Past that introductory level, in their junior years students of pure mathematics must be reintroduced to Calculus in a rigorous proof-driven way - thus enters the dreaded subject of "analysis", also sometimes called "advanced calculus" and thankfully, thus also enters this book. Bryant starts off assuming that rational numbers behave as we know from elementary school and then constructs the real numbers by adding a completeness axiom. From there he introduces the concept of limits and also the epsilon-delta technique in an accessible way before going on to the topics of differentiation and integration. Even though this is a mathematically rigorous book, the author manages to keep things interesting by introducing topics and theorems in bite-sized chunks. Basically, the book doesn't go beyond the analysis of calculus normally taught at the undergraduate level, but rather reintroduces it properly and puts it on the rigorous plane with which all graduate mathematics students shall become familiar. Along with all that, there's an excellent selection of interesting exercises with solutions at the back. These exercises range from the rather simple to the very tricky. If you are a mathematics major, you will probably not be lucky enough to have this as your textbook in analysis class, but you should buy a copy and read it before and during the class so that you know what is really trying to be conveyed.

Great Introduction

This is a text for Real Analysis at the Junior Level (American university level). It goes to extreme lengths to make analysis understandable to people who have no prior exposure. The organization is good. Completeness is introduced early as (the "piggy in the middle"). Proofs are written in detail with fill-in-the-blank spots to force the reader to follow the argument. It has good exercises making it an easy book to teach out of. Excellent for the absolute beginner. Good candidate for the classroom.

Exposes Mathematical Analysis Without Set Theory Background

Mathematical analysis is a refinement of calculus, and a pathway into further branches of mathematics, including topology and advanced topics in algebra. Analysis, however, may not seem to be at all related to calculus at its initial stages. An introductory course on analysis can render an unprepared student, even with experience in other branches of mathematics, perplexed and challenged to an extreme. Only later in the analysis course are even the most basic topics of calculus introduced.One of the most important considerations prior to taking an analysis course is the level of background and understanding of mathematical logic. Set theory, a branch of mathematical logic, is in fact the basis of calculus as well. Due to an emphasis upon computations, however, the highest grades in calculus are possible without understanding, or even knowing of, this underlying foundation.This work is unique among those introducing analysis, in that it does not require a background in set theory. It in fact teaches numerous fundamental concepts of set theory, without stating that it is doing so. Examples provided are based on daily concrete experience, yet are altered for purposes of mathematical instruction. These descriptions are sufficiently general as to prepare the reader for when formal set theory is introduced in more rigorous textbooks.In addition to being an extremely readable and accessible work, solutions and hints are provided for every review question for every section of the book. This is in stark contrast to textbooks on the subject, which, while costing several times more, are typically designed for a classroom setting, and so leave all questions unanswered. This self-testing of the understanding of each section is crucial for subject matter requiring such attention to detail and precision.The numerous illustrations throughout the book are rendered clearly and with instructional purpose, yet are often drawn by hand, adding to the sense of familiarity with the author. All of the basic subject matter for a course on analysis is provided, yet has been specifically tailored for a reader in the stages of preparation, of review after completion, or one who is simply inquisitive as to what is required to comprehend analysis successfully.The softcover edition is durable and portable, and the book remains in excellent condition through numerous readings, which it will almost certainly go through.If you have been required to take an analysis class but left it with only a vague sense of its underpinnings, you may wish to go through this work when time permits. For the price of the book, the information and instruction provided is truly outstanding. This text receives the highest marks in all categories.

Outstanding introduction to advanced mathematics

While there have been countless introductions to mathematical analysis (calculus) this is my favorite. The author does a brilliant job of making the subject matter interesting and very understandable with excellent exercises along the way which have solutions in the back ! A must read for bright highschool seniors and college freshman that are taking calculus or will be.