Introduction to Analysis

Maxwell Rosenlicht
Publisher:
Dover Publications
Publication Date:
1985
Number of Pages:
254
Format:
Paperback
Price:
14.95
ISBN:
9780486650388
Category:
Textbook
BLL Rating:

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on
07/16/2015
]

This can be thought of either as a brief introduction to real analysis, or as a rigorous calculus book: it proves nearly all the facts that are used in single- and multi-variable calculus, but generally does not go beyond that to the more general problems considered in real analysis. This is a 1986 Dover unaltered reprint of the 1968 edition from Scott, Foresman.

The book does cover some topics that are usually not touched on in calculus, such as some theory of complete metric spaces, uniform convergence and uniform continuity, and the inverse and implicit function theorems. Unlike most calculus courses, everything is done analytically rather than geometrically; for example, the transcendental functions are defined by power series. The book does not construct the real numbers but develops them axiomatically. There is an introductory chapter on set theory, but set theory is used in the book primarily for notation rather than as a foundation.

The exercises are excellent; there are lots of them, they cover many important additional results, and they have a wide range of difficulty. Usually each problem set starts out with exercises to find some examples of something, and then switches over to proving things. There are no answers or hints given for the exercises, and there aren’t many worked examples, so the audience is probably upper-division undergraduates.

A more recent book that is similar in many ways is Ross’s Elementary Analysis: The Theory of Calculus. Ross sticks more closely to the single-variable calculus syllabus and has few excursions into more advanced topics and nothing on multi-variable calculus. Ross’s book is probably better for present-day students, because it has many more worked examples, and it has (brief) answers to many of the exercises in the back of the book. Both are very good books, but differ in scope and in the amount of hand-holding given. Both books are much more elementary than traditional real-analysis texts such as Rudin’s Principles of Real Analysis and Apostol’s Mathematical Analysis.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

• CHAPTER I. NOTIONS FROM SET THEORY
• 1. Sets and elements. Subsets
• 2. Operations on sets
• 3. Functions
• 4. Finite and infinite sets
• Problems
• CHAPTER II. THE REAL NUMBER SYSTEM
• 1. The field properties
• 2. Order
• 3. The least upper bound property
• 4. The existence of square roots
• Problems
• CHAPTER III. METRIC SPACES
• 1. Definition of metric space. Examples
• 2. Open and closed sets
• 3. Convergent sequences
• 4. Completeness
• 5. Compactness
• 6. Connectedness
• Problems
• CHAPTER IV. CONTINUOUS FUNCTIONS
• 1. Definition of continuity. Examples
• 2. Continuity and limits
• 3. The continuity of rational operations. Functions with values in $E^n$
• 4. Continuous functions on a compact metric space
• 5. Continuous functions on a connected metric space
• 6. Sequences of functions
• Problems
• CHAPTER V. DIFFERENTIATION
• 1. The definition of derivative
• 2. Rules of differentiation
• 3. The mean value theorem
• 4. Taylor's theorem
• Problems
• CHAPTER VI. RIEMANN INTEGRATION
• 1. Definitions and examples
• 2. Linearity and order properties of the integral
• 3. Existence of the integral
• 4. The fundamental theorem of calculus
• 5. The logarithmic and exponential functions
• Problems
• CHAPTER VII. INTERCHANGE OF LIMIT OPERATIONS
• 1. Integration and differentiation of sequences of functions
• 2. Infinite series
• 3. Power series
• 4. The trigonometric functions
• 5. Differentiation under the integral sign
• Problems
• CHAPTER VIII. THE METHOD OF SUCCESSIVE APPROXIMATIONS
• 1. The fixed point theorem
• 2. The simplest case of the implicit function theorem
• 3. Existence and uniqueness theorems for ordinary differential equations
• Problems
• CHAPTER IX. PARTIAL DIFFERENTIATION
• 1. Definitions and basic properties
• 2. Higher derivatives
• 3. The implicit function theorem
• Problems
• CHAPTER X. MULTIPLE INTEGRALS
• 1. Riemann integration on a closed interval in $E^n$. Examples and basic properties
• 2. Existence of the integral. Integration on arbitrary subsets of $E^n$. Volume
• 3. Iterated integrals
• 4. Change of variable
• Problems