Real Mathematical Analysis

Description

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. 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