Leonard F. Richardson 
Advanced Calculus 
An Introduction to Linear Analysis

Features an introduction to advanced calculus and highlights its
inherent concepts from linear algebra

Advanced Calculus reflects the unifying role of linear
algebra in an effort to smooth readers’ transition to advanced
mathematics. The book fosters the development of complete
theorem-proving skills through abundant exercises while also
promoting a sound approach to the study. The traditional theorems
of elementary differential and integral calculus are rigorously
established, presenting the foundations of calculus in a way that
reorients thinking toward modern analysis.

Following an introduction dedicated to writing proofs, the book
is divided into three parts:

Part One explores foundational one-variable calculus topics from
the viewpoint of linear spaces, norms, completeness, and linear
functionals.

Part Two covers Fourier series and Stieltjes integration, which
are advanced one-variable topics.

Part Three is dedicated to multivariable advanced calculus,
including inverse and implicit function theorems and Jacobian
theorems for multiple integrals.

Numerous exercises guide readers through the creation of their
own proofs, and they also put newly learned methods into practice.
In addition, a ‘Test Yourself’ section at the end of each chapter
consists of short questions that reinforce the understanding of
basic concepts and theorems. The answers to these questions and
other selected exercises can be found at the end of the book along
with an appendix that outlines key terms and symbols from set
theory.

Guiding readers from the study of the topology of the real line
to the beginning theorems and concepts of graduate analysis,
Advanced Calculus is an ideal text for courses in advanced
calculus and introductory analysis at the upper-undergraduate and
beginning-graduate levels. It also serves as a valuable reference
for engineers, scientists, and mathematicians.

Table of Content

Preface.

Acknowledgments.

Introduction.

PART I. ADVANCED CALCULUS IN ONE VARIABLE.

1. Real Numbers and Limits of Sequences.

2. Continuous Functions.

3. Rieman Integral.

4. The Derivative.

5. Infinite Series.

PART II. ADVANCED TOPICS IN ONE VARIABLE.

6. Fourier Series.

7. The Riemann-Stieltjes Integral.

PART III. ADVANCED CALCULUS IN SEVERAL VARIABLES.

8. Euclidean Space.

9. Continuous Functions on Euclidean Space.

10. The Derivative in Euclidean Space.

11. Riemann Integration in Euclidean Space.

Appendix A. Set Theory.

Problem Solutions.

References.

Index.

About the author

Leonard F. Richardson, Ph D, is Herbert Huey Mc Elveen Professor and Assistant Chair of the Department of Mathematics at Louisiana State University, where he is also Director of Graduate Studies in Mathematics. Dr. Richardson’s research interests include harmonic analysis and representation theory.