# 10 Best Real Analysis Books for Math Majors

Looking for the best Real Analysis books? If **yes**, then you are at the right place. In this article, I have listed and reviewed the top 10 real analysis books sorted by their popularity and content quality.

These books are not just the best real analysis books but these also create a solid foundation for you going ahead.

What makes me eligible to say which book is better and which one is not? Well, **(a)** I am a Maths Postgraduate, and **(b)** I have read all of these.

So, without further ado, let’s dive into the list of the best real analysis books ever written.

**10 Best Real Analysis Books**

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**1. Principles of Mathematical Analysis**

Author – Walter Rudin

Principles of Mathematical Analysis by Walter Rudin (math lovers call it **little Rudin**) is one of the most well-known and respected books on the subject. It seeks to help both undergraduate and first-year graduate students develop a solid foundation in mathematical analysis.

It starts by discussing the real number system as a complete ordered field; in the latest edition of this book, Dedekind’s Cut is described in an appendix to Chapter 1.

In Chapter 2, you will find the topological background required for the development of differentiation, integration, continuity, and convergence.

The latest edition also includes an all-new section describing the gamma function, along with several new and interesting exercises. It is an essential part of the Walter Rudin Student Series in Advanced mathematics.

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**2. Real Analysis by Carothers**

Author – N.L. Carothers

Real Analysis by N.L. Carothers is aimed at advanced undergraduates and beginning graduates in mathematics and other related fields. It requires the reader to have only a basic understanding of advanced calculus and real analysis. Thus, it is a great recommendation for both specialists and non-specialists in the field. The book covers three important topics in this subject:

- Function spaces
- Metric and normed linear spaces
- Lebesgue measure and integration on the line

Carothers’ writing style is enticing and informal. He provides an overview of new ideas and encourages readers to understand them while providing complete details and proofs alongside. The book also contains numerous helpful exercises and suggestions for further study.

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**3. Real Analysis by Royden**

Author – H.L. Royden

This book serves as an introductory graduate text which deals with Lebesque integration and measure theory. Analysis on the real number line as usually studied at the advanced undergraduate level constitutes only a preliminary understanding of a much larger domain. In this book, Royden encourages the reader to return to the subject and relearn it from the most advanced point of view.

The first half of the book deals with theorems on the real line, while the second half is dedicated to theorems on arbitrary topological spaces. The author’s writing style is charmingly simple. Instead of making things unnecessarily complicated, he keeps the text lucid, clear, and to the point. After finishing the book, the reader will have a decent exposure to real analysis at a fairly advanced level.

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**4. Counterexamples in Analysis by Gelbaum**

Author – Bernard R. Gelbaum

This compelling text by Bernard R. Gelbaum is a great choice for students of analysis. The counterexamples listed here largely deal with the part of analysis known as “real variables”. The first half of the book deals with the real number system, differentiation Riemann integration, functions and limits, infinite series, sequences, and much more. The second half describes plane sets, metric and topological spaces, area, function spaces, and functions of two variables.

The book offers thought-provoking counterexamples and illustrates why certain theorems are phrased the way they are. If you need to produce functions with rather bizarre properties, then this is a particularly good resource. It offers a concise collection of solid counterexamples, some of which might have gone unnoticed by many.

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**5. Real Analysis: Modern Techniques and Their Applications by Folland**

Author – Gerald B. Folland

Gerald B. Folland’s Real Analysis: Modern Techniques and Their Applications offers a comprehensive approach to real analysis and its applications. In this new edition, the subject is covered in even greater detail and at a much more advanced level than most books presently available. It covers numerous subjects that modern analysis comprises of and emphasizes point-set topology, measure and integration theory, and the fundamentals of functional analysis.

The book also demonstrates the use of the general theories and provides an introduction to other branches of analysis, including probability theory, distribution theory, and Fourier analysis. This new edition features additional content and is also useful to students interested in dynamic forms. With a large bibliography, several exercises, and a review chapter on metric spaces and sets, it is a great choice for students in graduate-level analysis courses.

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**6. Principles of Real Analysis by S. C. Malik**

Author – S. C. Malik

Principles of Real Analysis serves as a textbook for postgraduate students of Indian universities and a course in Real Analysis for honors. It deals with the theory right away and explains the fundamentals of the subject meticulously.

Beginning with an overview of the essential properties of rational numbers and using Dedekind’s form, it establishes the properties of real numbers. This forms a solid foundation for the subsequent chapters, including differentiation, indeterminate forms, continuity, and much more. The book is notable for its plethora of well-graded examples, some of which have been solved for the benefit of the readers.

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**7. Introduction to Real Analysis by Bartle**

Author – Robert G. Bartle

In the modern era, mathematics has become an inseparable part of various disciplines – everything from management and economics to the physical sciences. This brilliant book by Robert G. Bartle presents the basic techniques and concepts of real analysis for readers in all of these disciplines. It encourages the reader to think rationally, analyze mathematical ideas, and think out of the box.

This new edition maintains the user-friendly approach and engaging nature of the first two editions. It also features some streamlined arguments, rearranged topics, new examples, and an all-new chapter on the Generalized Riemann Integral.

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**8. Real and Complex Analysis by Rudin**

Author – Walter Rudin

Real and Complex Analysis is a part of the Walter Rudin Student Series in Advanced Mathematics. It serves as an advanced text for the one- or two-semester course in analysis aimed at science, math, electrical engineering, and computer science majors at the junior, senior, and graduate levels. The book presents the fundamental theorems and techniques of analysis in a way that emphasizes the profound connections between its different branches.

The book is also notable for uniting the traditionally separate subjects of “complex analysis” and “real analysis” in one volume. It also presents some of the fundamental ideas from functional analysis. It has been arranged in a manner that allows every chapter to build upon the other. This allows students to gain a gradual and wholesome understanding of the subject.

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**9. Introductory Real Analysis by Kolmogorov**

Author – A.N. Kolmogorov

This is a part of Richard Silverman’s legendary series of translations of Russian works in mathematical science. It is a detailed and elementary introduction to real and functional analysis by two renowned faculty members from Moscow University. The book is self-contained, properly paced, simple to read, and easily accessible to readers who are well-versed in advanced calculus.

There are a total of 37 sections, with each of them containing a problem set, totaling up to around 350 meticulously chosen problems in all. This is a basic one-year course in real analysis and an excellent resource both for the classroom and self-study.

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**10. A First Course in Mathematical Analysis by Brannan**

Author – David Alexander Brannan

Students often consider mathematical analysis to be one of the toughest mathematics courses. This book by David Alexander Brannan aims to make the subject easily understandable by taking a sequential approach to continuity, differentiability, and integration. It lays strong emphasis on several topics that are usually glossed over in most standard calculus courses.

The text also features useful margin notes and numerous attractive diagrams. It guides students through some of the trickier points with the help of numerous graded exercises and examples. It is a great resource for both self-study and as a companion to any standard university course on the subject.

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**Summary**

Above books and their pricing can be summarized in this table:

**Also see:**

- Best Calculus Textbooks
- Best Free Online Calculus Books
- Best Free Online Algebra and Topology Books
- Best Books on Measure Theory

Last update on 2021-07-24 using Amazon Product Advertising API

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