Introduction to Real Analysis (Graduate Texts in Mathematics)

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