Based on an honors course taught by the author at UC Berkeley, this introduction to undergraduate real analysis gives a different emphasis by stressing the importance of pictures and hard problems. Topics include: a natural construction of the real numbers, four-dimensional visualization, basic point-set topology, function spaces, multivariable calculus via differential forms (leading to a simple proof of the Brouwer Fixed Point Theorem), and a pictorial treatment of Lebesgue theory. Over 150 detailed illustrations elucidate abstract concepts and salient points in proofs. The exposition is informal and relaxed, with many helpful asides, examples, some jokes, and occasional comments from mathematicians, such as Littlewood, Dieudonné, and Osserman. This book thus succeeds in being more comprehensive, more comprehensible, and more enjoyable, than standard introductions to analysis.
New to the second edition of Real Mathematical Analysis is a presentation of Lebesgue integration done almost entirely using the undergraph approach of Burkill. Payoffs include: concise picture proofs of the Monotone and Dominated Convergence Theorems, a one-line/one-picture proof of Fubini's theorem from Cavalieri’s Principle, and, in many cases, the ability to see an integral result from measure theory. The presentation includes Vitali’s Covering Lemma, density points — which are rarely treated in books at this level — and the almost everywhere differentiability of monotone functions. Several new exercises now join a collection of over 500 exercises that pose interesting challenges and introduce special topics to the student keen on mastering this beautiful subject.
Author(s): Charles Chapman Pugh
Edition: 2
Publisher: Springer
Year: 2017
Language: English
Commentary: corrected 2nd printing 2017
Pages: 486
Tags: Real Analysis; Mathematics; Measure and Integration; Real Functions; Sequences, Series, Summability
Preface
Contents
1 Real Numbers
1 Preliminaries
Language
Truth
Logic
Metaphor and Analogy
Two Pieces of Advice
2 Cuts
Cauchy sequences
Further description of R
The є-principle
3 Euclidean Space
Inner product spaces
4 Cardinality
5* Comparing Cardinalities
6* The Skeleton of Calculus
7* Visualizing the Fourth Dimension
Exercises
2 A Taste of Topology
1 Metric Spaces
Convergent Sequences and Subsequences
2 Continuity
Homeomorphism
The ( є, δ)-Condition
3 The Topology of a Metric Space
Topological Description of Continuity
Inheritance
Product Metrics
Completeness
4 Compactness
Ten Examples of Compact Sets
Nests of Compacts
Continuity and Compactness
Homeomorphisms and Compactness
Embedding a Compact
Uniform Continuity and Compactness
5 Connectedness
6 Other Metric Space Concepts
Clustering and Condensing
Perfect Metric Spaces
Continuity of Arithmetic in R
Boundedness
7 Coverings
Total Boundedness
8 Cantor Sets
9* Cantor Set Lore
Peano Curves
Cantor Spaces
Ambient Topological Equivalence
Antoine’s Necklace
10* Completion
A Second Construction of R from Q
Exercises
Prelim Problems†
3 Functions of a Real Variable
1 Differentiation
Pathological Examples
Higher Derivatives
Smoothness Classes
Analytic Functions
A Nonanalytic Smooth Function
Taylor Approximation
Inverse Functions
2 Riemann Integration
Darboux Integrability
Improper Integrals
3 Series
Conditional Convergence
Series of Functions
Exercises
4 Function Spaces
1 Uniform Convergence and C0[a, b]
2 Power Series
3 Compactness and Equicontinuity in C0
4 Uniform Approximation in C0
5 Contractions and ODEs
Ordinary Differential Equations
6* Analytic Functions
7* Nowhere Differentiable Continuous Functions
8* Spaces of Unbounded Functions
Exercises
More Prelim Problems
5 Multivariable Calculus
1 Linear Algebra
2 Derivatives
3 Higher Derivatives
Smoothness Classes
4 Implicit and Inverse Functions
5* The Rank Theorem
6* Lagrange Multipliers
7 Multiple Integrals
8 Differential Forms
Form Naturality
Form Names
Wedge Products
The Exterior Derivative
Pushforward and Pullback
9 The General Stokes Formula
Stokes’ Formula on Manifolds
Vector Calculus
Closed Forms and Exact Forms
Cohomology
Differential Forms Viewed Pointwise
10* The Brouwer Fixed-Point Theorem
Appendix A Perorations of Dieudonné
Appendix B The History of Cavalieri’s Principle
Appendix C A Short Excursion into the Complex Field
Appendix D Polar Form
Appendix E Determinants
Exercises
6 Lebesgue Theory
1 Outer Measure
2 Measurability
3 Meseomorphism
Affine Motions
4 Regularity
Inner Measure, Hulls, and Kernels
5 Products and Slices
6 Lebesgue Integrals
7 Italian Measure Theory
8 Vitali Coverings and Density Points
Density Points
9 Calculus à la Lebesgue
10 Lebesgue’s Last Theorem
Appendix A Lebesgue integrals as limits
Appendix B Nonmeasurable sets
Appendix C Borel versus Lebesgue
Appendix D The Banach-Tarski Paradox
Appendix E Riemann integrals as undergraphs
Appendix F Littlewood’s Three Principles
Appendix G Roundness
Appendix H Money
Exercises
Suggested Reading
Bibliography
Index
