A Course in Mathematical Analysis

by D.J.H. Garling

Three volumes that provide a full and detailed account of all those elements of real and complex analysis an undergraduate mathematics student can expect to encounter in the first two or three years of study. Numerous exercises, examples and applications are included.

D. J. H. Garling is Emeritus Reader in Mathematical Analysis at the University of Cambridge and Fellow of St John's College, Cambridge. He has fifty years' experience of teaching undergraduate students in most areas of pure mathematics, but particularly in analysis.

Contents

Introduction
Part One - Prologue: The foundations of analysis
1 - The axioms of set theory
2 - Number systems
Part Two - Functions of a real variable
3 - Convergent sequences
4 - Infinite series
5 - The topology of R
6 - Continuity
7 - Differentiation
8 - Integration
9 - Introduction to Fourier series
10 - Some applications
Appendix A - Zorn's lemma and the well-ordering principle
Index

Introduction
Part Three - Metric and topological spaces
11 - Metric spaces and normed spaces
12 - Convergence, continuity and topology
13 - Topological spaces
14 - Completeness
15 - Compactness
16 - Connectedness
Part Four - Functions of a vector variable
17 - Differentiating functions of a vector variable
18 - Integrating functions of several variables
19 - Differential manifolds in Euclidean space
Appendix B - Linear Algebra
Appendix C - Exterior algebras and the cross product
Appendix D - Tychonoff's theorem
Index

Introduction
Part Five - Complex Analysis
20 - Holomorphic functions and analytic functions
21 - The topology of the complex plane
22 - Complex integration
23 - Zeros and singularities
24 - The calculus of residues
25 - Conformal transformations
26 - Applications
Part Six - Measure and Integration
27 - Lebesgue measure on R
28 - Measurable spaces and measurable functions
29 - Integration
30 - Constructing measures
31 - Signed measures and complex measures
32 - Measures on metric spaces
33 - Differentiation
34 - Applications
Index

Genres: Mathematics

986 pages, Hardcover

Published July 24, 2014

Reviews

[Chiaki · Rating 3 out of 5]

There is no shortage of quality mathematical textbooks on analysis. So why write another? Garling is a stalwart supporter of the traditional approach to teaching mathematics emblematic of the Mathematical Tripos at the University of Cambridge; indeed, he claims to have over fifty years experience teaching undergraduates, mostly while at St. John's College, the great rival in mathematics teaching and next-door neighbour to the famous Trinity College.

So for Americans, read "first two or three years of study" to mean this book is suitable for advanced undergraduates and beginning graduates. If you enjoy the austere beauty of such authors as Rudin and Lang, and are looking for a new undergraduate textbook for your next course in real analysis, metric spaces, or measure theory, this might be the book for you. For everyone else: proceed with caution.

[henry · Rating 5 out of 5]

I feel like I have been waiting for this book all my life.