A Course in Mathematical Analysis: Foundations and Elementary Real Analysis by D. J. H. Garling – A Comprehensive Underg

Introduction

A Course in Mathematical Analysis: Foundations and Elementary Real Analysis is the first volume of a celebrated three-part series by D. J. H. Garling, published by Cambridge University Press. Designed for undergraduate mathematics students, this hardbound edition provides a rigorous yet accessible introduction to real analysis. For Indian students pursuing B.Sc., B.A. (Hons.) Mathematics, or preparing for competitive exams like the IIT JAM, this book is an indispensable resource that builds a solid foundation in mathematical thinking.

Book Overview

This volume focuses exclusively on the analysis of real-valued functions of a real variable. The author begins with the axioms of set theory, constructing the real number system from the ground up. The book then progresses through limits, continuity, differentiation, integration, and sequences and series of functions. A standout feature is a dedicated chapter on Fourier series, which bridges pure analysis with practical applications. Hundreds of exercises, examples, and applications are woven into the narrative, making abstract concepts tangible.

Key Highlights

• Foundational Approach: Starts with set theory axioms and constructs real numbers, ensuring no gaps in understanding.
• Rigorous Yet Clear: Every theorem is proved step by step, with careful explanations suitable for self-study.
• Extensive Exercises: Over 400 problems, ranging from routine drills to challenging proofs, with hints for selected ones.
• Applications-Focused: Includes Fourier series, Taylor expansions, and the Riemann–Stieltjes integral to connect theory with real-world use.
• Indian Curriculum Alignment: Covers topics typical in first- and second-year B.Sc./B.A. Mathematics programs across Indian universities.

Inside the Book

The content is organized into four parts: Prologue (set theory and construction of real numbers), Functions of a Real Variable (limits, continuity, differentiation, Riemann integration), Sequences and Series (uniform convergence, power series), and Fourier Series. Each chapter ends with a rich collection of exercises. The book also includes appendices on logic, sets, and functions for quick reference.

Key Topics

• Set Theory and Real Numbers: Zermelo–Fraenkel axioms, Dedekind cuts, completeness property
• Limits and Continuity: Epsilon-delta definitions, intermediate value theorem, uniform continuity
• Differentiation: Mean value theorems, L’Hôpital’s rule, Taylor’s theorem with remainder
• Riemann Integration: Darboux sums, fundamental theorem of calculus, improper integrals
• Sequences and Series: Convergence tests, power series, uniform convergence and its implications
• Fourier Series: Orthogonal systems, convergence theorems, applications to periodic functions

Reader Benefits

• Builds Mathematical Maturity: Teaches how to read, write, and construct rigorous proofs—a skill essential for higher studies.
• Self-Study Friendly: Clear exposition, solved examples, and hints for exercises make it ideal for independent learners.
• Exam Preparation: Covers core topics for university exams, GATE Mathematics, and NET/JRF in Mathematical Sciences.
• Lifetime Reference: The foundational nature of the content means you will revisit this book throughout your academic career.

Learning Outcomes

By working through this book, the reader will be able to: (1) understand the logical construction of the real number system; (2) apply epsilon-delta arguments confidently; (3) prove fundamental theorems of calculus; (4) analyze convergence of sequences and series of functions; (5) compute Fourier series expansions and understand their convergence; and (6) develop the ability to write clear, rigorous mathematical proofs.

Who Should Read

• Undergraduate Mathematics Students: Especially those in their first or second year of B.Sc. or B.A. (Hons.) Mathematics.
• Self-Learners: Anyone with a basic knowledge of calculus who wants to delve into rigorous analysis.
• Competitive Exam Aspirants: Students preparing for IIT JAM Mathematics, GATE, CSIR NET, or NBHM.
• Teachers and Tutors: An excellent source for problem sets and lecture material for real analysis courses.

About the Author

D. J. H. Garling is a distinguished mathematician and former lecturer at the University of Cambridge. With decades of teaching experience, he has authored several influential textbooks on analysis. His writing style is known for its precision, clarity, and pedagogical insight, making complex ideas accessible to students.

About the Publisher

Cambridge University Press is one of the world’s oldest and most respected academic publishers. Established in 1534, it has a long tradition of producing high-quality mathematics texts. This volume is part of their renowned Cambridge Mathematical Textbooks series, trusted by universities globally.

Conclusion

A Course in Mathematical Analysis: Foundations and Elementary Real Analysis is more than just a textbook—it is a gateway to advanced mathematics. Whether you are a student in an Indian university, a self-learner, or a teacher looking for a reliable reference, this hardcover edition from Cambridge University Press will serve you for years. Its rigorous yet supportive approach ensures that you not only learn analysis but truly understand it.