Introduction to Real Analysis (Graduate Texts in Mathematics)

✍ Scribed by Christopher Heil

• Publisher: Springer
• Year: 2019
• Tongue: English
• Leaves: 416
• Series: Graduate Texts in Mathematics (Book 280)
• Edition: 1st ed. 2019
• Category: Library

✦ Synopsis

Developed over years of classroom use, this textbook provides a clear and accessible approach to real analysis. This modern interpretation is based on the author’s lecture notes and has been meticulously tailored to motivate students and inspire readers to explore the material, and to continue exploring even after they have finished the book. The definitions, theorems, and proofs contained within are presented with mathematical rigor, but conveyed in an accessible manner and with language and motivation meant for students who have not taken a previous course on this subject.

The text covers all of the topics essential for an introductory course, including Lebesgue measure, measurable functions, Lebesgue integrals, differentiation, absolute continuity, Banach and Hilbert spaces, and more. Throughout each chapter, challenging exercises are presented, and the end of each section includes additional problems. Such an inclusive approach creates an abundance of opportunities for readers to develop their understanding, and aids instructors as they plan their coursework. Additional resources are available online, including expanded chapters, enrichment exercises, a detailed course outline, and much more.

Introduction to Real Analysis is intended for first-year graduate students taking a first course in real analysis, as well as for instructors seeking detailed lecture material with structure and accessibility in mind. Additionally, its content is appropriate for Ph.D. students in any scientific or engineering discipline who have taken a standard upper-level undergraduate real analysis course.

✦ Table of Contents

Contents
Preface
Audience
Online Resources
Outline
Course Options
Acknowledgments
Preliminaries
Numbers
Sets
Equivalence Relations
Intervals
Euclidean Space
Sequences
The Kronecker Delta and the Standard Basis Vectors
Functions
Cardinality
Extended Real-Valued Functions
Notation for Extended Real-Valued and Complex-Valued Functions
Suprema and Infima
Convergent and Cauchy Sequences of Scalars
Convergence in the Extended Real Sense
Limsup and Liminf
Infinite Series
Pointwise Convergence of Functions
Continuity
Derivatives and Everywhere Differentiability
The Riemann Integral
1 Metric and Normed Spaces
1.1 Metric Spaces
1.1.1 Convergence and Completeness
1.1.2 Topology in Metric Spaces
1.1.3 Compact Sets in Metric Spaces
1.1.4 Continuity for Functions on Metric Spaces
Problems
1.2 Normed Spaces
1.2.1 Vector Spaces
1.2.2 Seminorms and Norms
1.2.3 Infinite Series in Normed Spaces
1.2.4 Equivalent Norms
Problems
1.3 The Uniform Norm
1.3.1 Some Function Spaces
Problems
1.4 Hölder and Lipschitz Continuity
Problems
2 Lebesgue Measure
2.1 Exterior Lebesgue Measure
2.1.1 Boxes
2.1.2 Some Facts about Boxes
2.1.3 Exterior Lebesgue Measure
2.1.4 The Exterior Measure of a Box
2.1.5 The Cantor Set
2.1.6 Regularity of Exterior Measure
Problems
2.2 Lebesgue Measure
2.2.1 Definition and Basic Properties
2.2.2 Toward Countable Additivity and Closure under Complements
2.2.3 Countable Additivity
2.2.4 Equivalent Formulations of Measurability
2.2.5 Carathéodory’s Criterion
2.2.6 Almost Everywhere and the Essential Supremum
Problems
2.3 More Properties of Lebesgue Measure
2.3.1 Continuity from Above and Below
2.3.2 Cartesian Products
2.3.3 Linear Changes of Variable
Problems
2.4 Nonmeasurable Sets
2.4.1 The Axiom of Choice
2.4.2 Existence of a Nonmeasurable Set
2.4.3 Further Results
Problems
3 Measurable Functions
3.1 Definition and Properties of Measurable Functions
3.1.1 Extended Real-Valued Functions
3.1.2 Complex-Valued Functions
Problems
3.2 Operations on Functions
3.2.1 Sums and Products
3.2.2 Compositions
3.2.3 Suprema and Limits
3.2.4 Simple Functions
Problems
3.3 The Lebesgue Space L∞(E)
3.3.1 Convergence and Completeness in L∞(E)
Problems
3.4 Egorov’s Theorem
Problems
3.5 Convergence in Measure
Problems
3.6 Luzin’s Theorem
Problems
4 The Lebesgue Integral
4.1 The Lebesgue Integral of Nonnegative Functions
4.1.1 Integration of Nonnegative Simple Functions
4.1.2 Integration of Nonnegative Functions
Problems
4.2 The Monotone Convergence Theorem and Fatou’s Lemma
4.2.1 The Monotone Convergence Theorem
4.2.2 Fatou’s Lemma
Problems
4.3 The Lebesgue Integral of Measurable Functions
4.3.1 Extended Real-Valued Functions
4.3.2 Complex-Valued Functions
4.3.3 Properties of the Integral
Problems
4.4 Integrable Functions and L1(E)
4.4.1 The Lebesgue Space L1(E)
4.4.2 Convergence in L1 -Norm
4.4.3 Linearity of the Integral for Integrable Functions
4.4.4 Inclusions between L1(E) and L1(E)
Problems
4.5 The Dominated Convergence Theorem
4.5.1 The Dominated Convergence Theorem
4.5.2 First Applications of the DCT
4.5.3 Approximation by Continuous Functions
4.5.4 Approximation by Really Simple Functions
4.5.5 Relation to the Riemann Integral
Problems
4.6 Repeated Integration
4.6.1 Fubini’s Theorem
4.6.2 Tonelli’s Theorem
4.6.3 Convolution
Problems
5 Differentiation
5.1 The Cantor–Lebesgue Function
Problems
5.2 Functions of Bounded Variation
5.2.1 Definition and Examples
5.2.2 Lipschitz and Hölder Continuous Functions
5.2.3 Indefinite Integrals and Antiderivatives
5.2.4 The Jordan Decomposition
Problems
5.3 Covering Lemmas
5.3.1 The Simple Vitali Lemma
5.3.2 The Vitali Covering Lemma
Problems
5.4 Differentiability of Monotone Functions
Problems
5.5 The Lebesgue Differentiation Theorem
5.5.1 L1-Convergence of Averages
5.5.2 Locally Integrable Functions
5.5.3 The Maximal Theorem
5.5.4 The Lebesgue Differentiation Theorem
5.5.5 Lebesgue Points
Problems
6 Absolute Continuity and the Fundamental Theorem of Calculus
6.1 Absolutely Continuous Functions
6.1.1 Differentiability of Absolutely Continuous Functions
Problems
6.2 Growth Lemmas
Problems
6.3 The Banach–Zaretsky Theorem
Problems
6.4 The Fundamental Theorem of Calculus
6.4.1 Applications of the FTC
6.4.2 Integration by Parts
Problems
6.5 The Chain Rule and Changes of Variable
Problems
6.6 Convex Functions and Jensen’s Inequality
Problems
7 The Lp Spaces
7.1 The ℓp Spaces
7.1.1 Hölder’s Inequality
7.1.2 Minkowski’s Inequality
7.1.3 Convergence in the ℓp Spaces
7.1.4 Completeness of the ℓp Spaces
7.1.5 ℓp for p < 1
7.2 The Lebesgue Space Lp(E)
7.2.1 Seminorm Properties of || • ||p
7.2.2 Identifying Functions That Are Equal Almost Everywhere
7.2.3 Lp(E) for 0 < p < 1
7.2.4 The Converse of Hölder’s Inequality
Problems
7.3 Convergence in Lp-norm
7.3.1 Dense Subsets of Lp(E)
7.4 Separability of Lp(E)
Problems
8 Hilbert Spaces and L2(E)
8.1 Inner Products and Hilbert Spaces
8.1.1 The Definition of an Inner Product
8.1.2 Properties of an Inner Product
8.1.3 Hilbert Spaces
Problems
8.2 Orthogonality
8.2.1 Orthogonal Complements
8.2.2 Orthogonal Projections
8.2.3 Characterizations of the Orthogonal Projection
8.2.4 The Closed Span
8.2.5 The Complement of the Complement
8.2.6 Complete Sequences
Problems
8.3 Orthonormal Sequences and Orthonormal Bases
8.3.1 Orthonormal Sequences
8.3.2 Unconditional Convergence
8.3.3 Orthogonal Projections Revisited
8.3.4 Orthonormal Bases
8.3.5 Existence of an Orthonormal Basis
8.3.6 The Legendre Polynomials
8.3.7 The Haar System
8.3.8 Unitary Operators
Problems
8.4 The Trigonometric System
Problems
9 Convolution and the Fourier Transform
9.1 Convolution
9.1.1 The Definition of Convolution
9.1.2 Existence
9.1.3 Convolution as Averaging
9.1.4 Approximate Identities
9.1.5 Young’s Inequality
Problems
9.2 The Fourier Transform
9.2.1 The Inversion Formula
9.2.2 Smoothness and Decay
Problems
9.3 Fourier Series
9.3.1 Periodic Functions
9.3.2 Decay of Fourier Coefficients
9.3.3 Convolution of Periodic Functions
9.3.4 Approximate Identities and the Inversion Formula
9.3.5 Completeness of the Trigonometric System
9.3.6 Convergence of Fourier Series for for p≠2
Problems
9.4 The Fourier Transform on on L2(R)
Problems
Hints for Selected Exercises and Problems
Index of Symbols
References
Index