Real analysis

Preface Contents Introduction Calculus and Analysis Differential Calculus Measure and Integration Measure Integration Other Topics What is Mathematics, and Why the Bad Reputation? Rigor in Mathematics Semantics: What is Obvious? Set Theory and Logic Proofs vs. Formal Proofs (or, How Mathematics Really Works) How to Understand a Proof Proof by Contradiction Proofs and Counterexamples: Mathematics as Creative Activity Scholarship vs. Practice Is Mathematics Created or Discovered? Analogies for the Structure of Mathematics Pure vs. Applied Mathematics Is Mathematics an Art or a Science? The Foundations of Mathematics Philosophies of Mathematics Mathematical Theories Languages Logic Mathematical Structures and Models Printed or Online? How Much Generality? History vs. Logic The Formalities of Mathematics Formal Statements, Negations, and Mathematical Arguments Formal Statements Meaning of Statements Quantification and Closed Statements Negations Uniqueness Statements Tautologies Definitions Languages and Relational Systems Mathematical Arguments and Formal Proofs Exercises Set Theory New Sets from Old Relations and Functions General Relations Functions Equivalence Relations Exercises Ordered Sets and Induction Partially Ordered Sets Well-Ordered Sets and Transfinite Induction Exercises Directed Sets Exercises Cardinality of Sets Cardinality Finite and Infinite Sets Countable and Uncountable Sets Properties of Countable Sets Additional Facts about Cardinality Exercises The Axiom of Choice and Zorn's Lemma The Axiom of Choice Equivalent Formulations Other Forms of Choice Exercises Axiomatic Set Theory Ordinals Basic Definitions Ordering on Ordinals Transfinite Induction Ordinals as Universal Models The Axiom of Choice and Zorn's Lemma Ordinal Arithmetic and Examples Exercises Cardinals Cardinals as Ordinals General Cardinals and Alephs Cardinal Arithmetic The Role of the Axiom of Choice The Continuum Hypothesis Large Cardinals Exercises Measurable Cardinals Introduction Real-Valued Measurable Cardinals Atomless Full Real-Valued Measures Measurable Cardinals Consistency and Equiconsistency Results Categories and Functors Mathematical Theories Categories Functors Natural Transformations Equivalence of Categories and Skeletons Adjoint Functors The Category of Small Categories Categories as a Foundation of Mathematics Exercises The Numbers of Analysis The Real Numbers – Axiomatic Approach The Algebraic Axioms of R The Order Axioms for R Absolute Value The Natural Numbers Induction and Recursion The Integers The Rational Numbers The Number Line Suprema, Infima, and the Completeness Axiom The Archimedean Property and Density of Q Uniqueness of R Existence of Roots The Extended Real Line Exercises The Natural Numbers Introduction Axiomatic Characterization of N Induction and Recursion Algebraic Structure of N Algebraic Axiomatization of N Order Structure of N Subtraction and Division in N Axiomatization of N0 Further Properties of N Summary Exercises The Integers Introduction and Axiomatization of Z Models for Z Uniqueness of Z Ordering and Subtraction in Z Multiplication and Division in Z Ring Axiomatization of Z Exercises The Rational Numbers Introduction and Axiomatization of Q The Standard Model of Q Exercises The Real Numbers Construction of R: the Dedekind Cut Approach Construction of R: the Cauchy Sequence Approach Exercises The Complex Numbers Axiomatization of C The Standard Model for C: the Complex Plane Exercises Fundamental Theorems About R Partitions Compactness and Continuity The Creeping Lemma Exercises Representation of Real Numbers Decimal Expansions Continued Fractions Measure Theory of Continued Fractions Exercises Irrational and Transcendental Numbers Introduction Irrationality of Transcendence of e and Related Numbers Twentieth Century Results Exercises Hyperreal Numbers and Nonstandard Analysis Ordered Fields Containing R The Hyperreal Numbers via Ultrafilters Internal and External Sets and Functions The Transfer Principle Other Nonstandard Structures Proofs and Applications of Nonstandard Analysis Loeb Measures and a Nonstandard Approach to Measure Theory Exercises Linear Algebra Matrix Algebra Vector Spaces Linear Transformations The Matrix of a Linear Transformation Linear Functionals and Dual Spaces Covariant vs. Contravariant More on Matrices and Linear Equations Affine Subspaces and Affine Functions The Dot Product on Rn and Adjoints Determinants Eigenvalues and Eigenvectors Exercises Multilinear Algebra Tensor Products Modules The Tensor Algebra of a Vector Space The Exterior Algebra of a Vector Space Mixed Tensors and General Tensor Algebra Homological Algebra Exercises The Banach-Tarski Paradox Rigid Motions Rigid Equivalence Equivalence by Dissection Equidecomposability Free Groups and Paradoxical Decompositions The Hausdorff Paradox The Banach-Tarski Paradox, Basic Version The Banach-Tarski Paradox, General Version Exercises Sequences and Series Sequences Sequences of Numbers Subsequences and Tails Limits of Sequences Infinite Limits Monotone Sequences Subsequences and Cluster Points of Sequences Limit Superior and Limit Inferior Cauchy Sequences Sequences in Rn Sequences in Metric Spaces Exercises Infinite Series Partial Sums and Convergence Nonnegative Series Tests for Convergence of Nonnegative Series More Delicate Tests Absolute and Conditional Convergence Hypergeometric Series Scaling of Series Subseries and Rearrangements of Series Unordered Summations Double Summations Products of Infinite Series Infinite Products Infinite Series of Complex Numbers Numerical Evaluation and Estimation of Series Summability Methods for Divergent Series Exercises Convergence of Sequences of Functions Pointwise and Uniform Convergence U.C. Convergence Mean and Mean-Square Convergence Exercises Calculus Limits of Functions Ordinary Limits The Sequential Criterion Limit Theorems Indeterminate Forms One-Sided Limits Extended Limits Limit Superior and Limit Inferior Exercises Continuity on R Continuity at a Point Continuity on a Set The Intermediate Value Theorem The Max-Min Theorem Continuity and Oscillation Differentiation and Derivatives The Derivative Elementary Rules of Differentiation The Product and Quotient Rules The Chain Rule Derivatives of Inverse Functions Critical Points and Local Extrema Higher Derivatives Continuous but Nondifferentiable Functions The Bush-Wunderlich-Swift Example Exercises Calculus of Finite Differences Exercises Extensions of the Derivative One-Sided Derivatives Dini Derivatives Symmetric First and Second Derivatives Approximate and Weak Derivatives Exercises Exponential and Trigonometric Functions Rational Exponents Irrational Exponents and Exponential Functions The Number e Derivatives of Exponential and Logarithm Functions Exercises Trigonometric Functions Arc Length of a Circle and the Arcsine Function The Sine and Cosine Functions Other Trigonometric Functions The Inverse Trigonometric Functions The Area of a Circle The Hyperbolic Functions Exercises Big-O and Little-o Notation Exercises The Mean Value Theorem Rolle's Theorem and the Mean Value Theorem Applications of the Mean Value Theorem Geometric Interpretation of the Second Derivative Refinements of the Mean Value Theorem Antiderivatives and Primitives Exercises L'Hôpital's Rule L'Hôpital's Rule, Version 1 L'Hôpital's Rule, Version 2 Applications Exercises Taylor's Theorem Taylor Polynomials Vanishing Order Remainder Formulas The Integral Remainder Formula The Morse Lemma Applications Exercises Interpolation and Extrapolation The Interpolating Polynomial Linear Interpolation and Extrapolation Higher-Order Interpolation and Extrapolation The Mean Value Theorem for Divided Differences Repeated Points Osculating Polynomials How Good are the Approximations? Exercises Miscellaneous Topics in Calculus How Discontinuous Can a Derivative Be? Other Properties of Derivatives Analysis on Q Exercises The Binomial Theorem Exercises The Gamma Function Definition of the Gamma Function Log-Convexity and a Characterization of the Gamma Function Miscellaneous Additional Results Exercises Stirling's Formula Exercises The Volume of the Unit Ball in Rn Radial Borel Sets and Functions Calculation of n Surface Area of a Sphere Exercises Infinite Series of Functions Infinite Series of Functions Exercises Power Series Analytic Functions Definitions and Basic Properties Rigidity of Analytic Functions Combinations of Analytic Functions The Analytic Morse Lemma Smooth Non-Analytic Functions Analyticity at a Point (C)-Points and Smooth Partitions (P)-Points and Borel's Theorem Everywhere Nonanalytic Functions Exercises The Topology of R, Rn, and Metric Spaces The Topology of R Open and Closed Sets in R Limit Points and Closures Metric Spaces Definitions Balls, Limit Points, and Closures Limits and Continuity Extended Metrics Cauchy Sequences Normed Vector Spaces Exercises How Many Separable Metric Spaces Are There? The Banach Fixed-Point Theorem Exercises Euclidean Space Metric Axiomatization of Euclidean Space Rn is n-Dimensional Exercises Length Structures and Metric Geometry Arc Length in Metric Spaces Length Metrics Length Structures Metric Geometry Exercises Curves Curves and Parametrized Curves in Rn Parametrized Curves Reparametrization Paths Smooth and Piecewise Smooth Curves in Rn Smooth Functions from R to Rn Smooth Curves in Rn Exercises Rectifiable Curves and Arc Length Arc Length Rectifiable Curves Length of Arcs Integrating a Function Along a Curve Exercises Curves in R2 Curves in R3 Exercises Multivariable and Vector Calculus Limits and Continuity for Vector Functions Graphs of Functions of Several Variables Limits of Sequences in Rn Multivariable Limits Partial and Directional Derivatives Partial Derivatives Equality of Mixed Partials Directional Derivatives Exercises Differentiability of Multivariable Functions Differentiability and the Derivative Differentiability of Algebraic Combinations A Criterion for Differentiability The Chain Rule The Gradient and Tangent Planes Functional Independence Exercises Higher-Order Derivatives and Critical Point Analysis Higher-Order Differentiability Taylor's Theorem in Several Variables Critical Point Analysis The Morse Lemma Exercises The Implicit and Inverse Function Theorems Statements of the Theorems Twin Theorems and a Generalization Proof of the Implicit Function Theorem, Case n=1 Proof of the General Implicit Function Theorem Proof of the Inverse Function Theorem How Do We Find The Implicit/Inverse Function? How Much Can the Theorems be Generalized? The Domain and Range Straightening Theorems The Degenerate Case Analytic Versions Exercises Constrained Extrema and Lagrange Multipliers The Two-Dimensional Case The Higher-Dimensional Case with One Constraint The Higher-Dimensional Case with Multiple Constraints Degenerate Cases Exercises Power Series and Analytic Functions of Several Variables Multivariable Power Series Analytic Functions of Several Variables Differential Forms and Vector Analysis Vector Fields The Tangent and Cotangent Bundle on Rn Sections and Vector Fields Change of Basis Lie Differentiation Integral Curves The Local Flow Exercises Differential Forms Differential Forms on R1 Differential 1-Forms and Line Integrals on Rn Higher Differential Forms on Rn Integration of Differential Forms on Rn Stokes' Theorem in Rn Surfaces in R3 Divergence and Curl Analysis on Manifolds Exercises Differential Topology De Rham Cohomology Vector Bundles Topological K-Theory Differential Equations Existence and Uniqueness of Solutions to Ordinary Differential Equations Elementary Considerations and Terminology Existence and Uniqueness Theorem for First-Order ODEs Systems of First-Order Differential Equations Higher-Order Differential Equations and Systems Continuous Dependence on Initial Conditions Peano's Existence Theorem for First-Order ODEs Power Series and Analytic Solutions Linear ODEs Boundary-Value Problems Exercises Partial Differential Equations Elementary Theory Quasilinear Partial Differential Equations The Cauchy-Kovalevskaya Theorem Fundamentals of Complex Analysis Elementary Theory Complex Differentiation and Holomorphic Functions Rules of Differentiation Branch Cuts and Branches The Fundamental Theorem of Algebra Contour Integration Integration along a Contour Some Big Theorems Goursat's Theorem The Antiderivative Theorem and Cauchy's Theorem for Convex Sets The Cauchy Integral Formula The Homotopy Theorem for Contour Integrals Homology of Contours Cauchy's Theorem, General Versions The Cauchy Integral Formula, General Versions Sequences and Series of Holomorphic Functions; Analytic Functions Sequences and Series of Holomorphic Functions Power Series Inverse Power Series Analytic Functions Normal Families and Montel's Theorem Zeroes and Singularities Zeroes of Holomorphic Functions Singularities Behavior at Infinity Meromorphic Functions The Residue Theorem Principle of the Argument The Complex Morse Lemma Conformal Equivalence and the Riemann Mapping Theorem Conformal Equivalence Simply Connected Sets in C The Riemann Mapping Theorem Analytic Continuation and Riemann Surfaces Analytic Function Elements The Riemann Surface of a Total Analytic Function Analytic Continuation The Regular Riemann Surface of a Function Branch Points Points at Infinity and Meromorphic Functions Germs of Analytic Functions General Riemann Surfaces Exercises Analytic Functions of Several Variables Elementary Theory Exercises Differential Algebra Derivations Differential Fields Integration in Finite Terms Exercises Topology Topological Spaces Definitions Bases and Subbases Covers and Subcovers Relative Topology on a Subspace Separated Union Exercises Closed Sets and Limit Points Closed Sets Limit Points of a Set Closure of a Set Specification of a Topology via Closure Isolated Points and Derived Sets Exercises Convergence Limits of Sequences Nets and Limits Specification of a Topology via Convergence Convergence via Filters Exercises Continuous Functions Definition and Properties Topological Continuity vs. Metric Continuity Comparison of Topologies Stronger and Weaker Topologies Products and the Product Topology Exercises Separation Axioms The T-Axioms T0-Spaces T1-Spaces Hausdorff Spaces Regular Spaces Normal Spaces Completely Regular Spaces Exercises Quotient Spaces and the Quotient Topology The Quotient Topology Semicontinuous Decompositions Separation Axioms for Quotient Spaces Quotient Metrics Exercises Identification Spaces Collapsing a Subset to a Point Gluing and Attaching Group Actions on Spaces Local Homeomorphisms and Covering Maps Exercises Metrizable Spaces Metrizability Completely Metrizable Spaces A Universal Space Polish Spaces Exercises Compactness Basic Definitions Compactness and Convergence Compact Metrizable Spaces Separation Properties Automatic Continuity and Comparison of Topologies Products of Compact Spaces and Tikhonov's Theorem Variations on Compactness Locally Compact Spaces Compactifications Paracompactness and Partitions of Unity Compactly Generated Spaces Exercises Dini's Theorem Baire Spaces and the Baire Category Theorem Meager Sets and Baire Spaces The Baire Category Theorem Meager Sets vs. Null Sets The Baire Property Exercises Connectedness Connected Spaces Continuous Images of Connected Sets Unions and Intersections Components of a Space Products of Connected Spaces Totally Disconnected and Zero-Dimensional Spaces Locally Connected Spaces Path-Connected Spaces Exercises Direct and Inverse Limits Direct Systems and Direct Limits of Sets Inverse Systems and Inverse Limits of Sets Direct Limits in Topology Inverse Limits in Topology Exercises Uniform Structures Topologies and Uniform Structures Defined by Pseudometrics General Uniform Structures Order Topology Definition and Properties Completion of an Order Topology Function Space Notation and the Compact-Open Topology Exercises Cantor Sets The Cantor Set Other Cantor Sets Exercises The Topology of Rn Invariance of Domain and the Jordan Curve Theorem Rn is n-Dimensional When is Rn ``Similar'' to Rm? Compact Subsets of Rn Exercises Manifolds Topological Manifolds Topological Manifolds With Boundary New Manifolds From Old Classification of Topological Manifolds Exercises The Brouwer Fixed-Point Theorem Statements of the Results The Case n=2 The General Case Exercises Proof of the No-Retraction Theorem Topological Dimension Theory Essential Dimension Invariance of Domain Other Dimension Functions Comparing the Dimensions More Dimension Theories Embedding in Euclidean Space Infinite-Dimensional Spaces Exercises Homotopy and the Fundamental Group Homotopy The Fundamental Group Covering Spaces Other Homotopy Groups Exercises Fubini's Differentiation Theorem Exercises Sard's Theorem Sets of Measure Zero Statement of Sard's Theorem The Case n 1 The Case n>m, r Large The Case n>m, Sharp r Bound The Category Version Counterexamples Applications Refined Versions Exercises Piecewise-Linear Topology Polyhedra, Cells, and Simplexes Complexes Triangulations Piecewise-Linear Maps Exercises The Metric Space of a Measure Space Measurable Spaces and Subset Structures Algebras and -Algebras of Sets Set Operations Rings, Semirings, and Semialgebras Algebras and -Algebras -Rings and -Rings -Systems, -Systems, and Monotone Classes Measurable Spaces and Measurable Functions Exercises Filters and Ultrafilters Exercises Borel Sets and Borel Measurable Functions Borel Sets Real-Valued Measurable Functions Real-Valued Measurable Functions on -Rings Borel Measurable Functions Another Characterization of Borel Sets and Borel Measurable Functions Algebraic Properties of The Multiplicity Function Exercises Analytic Sets Standard Borel Spaces Analytic Sets Suslin Sets and the Suslin Operation Addison's Notation and the Projective Hierarchy Exercises Measure Theory Measures Definitions Elementary Properties Null Sets and Completeness Finite, -Finite, and Semifinite Measures Discrete and Continuous Measures Linear Combinations, Restrictions, and Extensions of Measures Measures Defined on Rings and -Rings Egorov's Theorem Measures on -Rings Exercises Probability Introduction The Dictionary Exercises Measures Defined by Extensions Outer Measures Lebesgue Outer Measure on R The Measure Induced by an Outer Measure Lebesgue Measure on R Extending Premeasures from Semirings The Suslin Operation Local Measures on -Rings and -Rings Radon Measures on R Lebesgue Measure and Radon Measures on Rn Nonmeasurable Subsets of R Lebesgue Measurable Functions Exercises Measures on Metric Spaces Metric Outer Measures and Borel Measures Method II Construction of Outer Measures Hausdorff Measure Hausdorff Measure on Rn Hausdorff Measure and Hausdorff Dimension for Self-Similar Sets Packing Measure and other Geometric Measures Radon Measures on Metric Spaces Exercises Measures on Product Spaces Product -Algebras Product Measures Lebesgue Measure on Rn Infinite Product Spaces Exercises Signed and Complex Measures Signed Measures Complex Measures Integration With Respect to Signed and Complex Measures Another Approach via Local Measures Exercises Signed and Complex Measures on -Rings Limits of Measures General Considerations Monotone and Dominated Convergence The Nikodým and Vitali-Hahn-Saks Theorems Exercises Integration Introduction Origin of Integration Theories The Cauchy Integral for Continuous Functions The Riemann Integral The Lebesgue Approach The Daniell Approach Direct Generalizations of Riemann Integration Oriented and Unoriented Integration The Riemann Integral Partitions, Tags, Riemann and Darboux Sums Step Functions and their Integrals Definitions of the Riemann Integral Comparing the Definitions Integrals of Continuous and Monotone Functions Properties of the Riemann Integral Null Sets and Almost Everywhere The Upper and Lower Envelope of a Riemann-Integrable Function Differentiation and the Fundamental Theorem of Calculus Integration by Substitution and Integration by Parts Improper Riemann Integration Exercises Numerical Integration and Simpson's Rule Exercises Jordan Content and Riemann Integration in Rn Integration over Rectangles Content and Jordan Regions Integration over Jordan Regions Fubini's Theorem Interchanging Differentiation and Integration Change of Variable Improper Integrals Step Functions The Monotone Convergence Theorem Characterization of Riemann Integrable Functions The Extended Riemann Integral Exercises The Lebesgue Integral Properties of the Integral Integral of Simple Functions The Integral of a Nonnegative Measurable Function The Monotone Convergence Theorem Consequences of the Monotone Convergence Theorem The Lebesgue Integral and Riemann Integral Continuity and Fatou's Lemma The Integral For General Functions Exercises The Dominated Convergence Theorem Applications of the DCT Uniform Integrability and Vitali's Convergence Theorem Exercises Integration on Product Spaces The Fubini-Tonelli Theorems Lebesgue Integration on Rn Exercises The Henstock-Kurzweil Integral Definition and Basic Properties The Fundamental Theorem of Calculus Convolution Convolution of Functions on R, Continuous Case The Radon-Nikodým Theorem Density Functions Statement and Proof of the Theorem The Lebesgue Decomposition Theorem Signed and Complex Measures Conditional Expectations Exercises Appendix B Space-Filling Curves The Borel Sets Are Not Countably Constructible Exercises A Plane Curve With Positive Area Borel Measures on R and Cumulative Distribution Functions Cumulative Distribution Functions Density Functions and Absolute Continuity Finite Borel Measures on Intervals Unbounded Distribution Functions and Positive Radon Measures Signed Measures and Nonincreasing Distribution Functions Monotone Functions Definitions and Basics Differentiability of Monotone Functions Fubini's Series Theorem Exercises Convex Functions Definitions Midpoint-Convex Functions Convergence of Sequences of Convex Functions Jensen's Inequality Log-Convex Functions Higher-Dimensional Versions of Convexity Exercises Functions of Bounded Variation The Total Variation of a Function The Total Variation Functions Banach's Multiplicity Theorem Extensions of Bounded Variation Exercises Absolute Continuity and Lebesgue Density Luzin's Property (N) Absolutely Continuous Functions Signed and Complex Borel Measures on R The Bounded Variation Norm Integration by Substitution and Integration by Parts Lp Versions, p>1 The Lebesgue Density Theorem Exercises Functional Analysis Normed Vector Spaces Completeness Isometries of Normed Vector Spaces Exercises Topological Vector Spaces Basic Definitions and Properties Locally Convex Topological Vector Spaces Metrizability and Normability Bounded Operators Bounded Operators on Normed Vector Spaces Bounded Linear Functionals Operators on Finite-Dimensional Normed Spaces Bounded Operators on Hilbert Spaces Algebras of Operators Dual Spaces and Weak Topologies Bounded Linear Functionals The Hahn-Banach Theorem Sublinear Functionals and Version 1 Extension of Bounded Linear Functionals The Complex Hahn-Banach Theorem Separation of Convex Sets Exercises The Open Mapping Theorem, Closed Graph Theorem, and Uniform Boundedness Theorem Open Mappings The Open Mapping Theorem The Closed Graph Theorem The Uniform Boundedness Theorem Exercises Compact Convex Sets Faces and Extreme Points Compact Convex Sets in Finite-Dimensional Spaces The Krein-Milman Theorem Choquet's Theorem Regular Embeddings, Cones, and Affine Functions Choquet Simplexes Exercises The Stone-Weierstrass Theorem The Weierstrass Approximation Theorem The Stone-Weierstrass Theorem Exercises Hilbert Spaces Inner Products The CBS Inequality and the Norm Completeness and Hilbert Spaces Best Approximation Orthogonality Projections Dual Spaces and Weak Topology Standard Constructions Real Hilbert Spaces Exercises Sesquilinear Forms and Operators Banach Algebras and Spectral Theory Banach Algebras Invertibility, Ideals, and Quotients Spectrum Holomorphic Functional Calculus Spectral Theory for Bounded Operators Exercises Fredholm Operators and Fredholm Index Banach Space Preliminaries Compact Operators Fredholm Operators Fredholm Operators of Index 0 and the Fredholm Alternative Relation between Compact and Fredholm Operators Exercises The Spectral Theorem The Finite-Dimensional Spectral Theorem Lp-Spaces Conjugate Exponents Lp-Spaces, Norms, and Inequalities Lp-Spaces L- and L-Spaces Comparison Between Lp-Spaces and Lp-Spaces Lp-spaces for Borel Measures The Dual of Lp-Spaces Conditional Expectations on Lp-Spaces Exercises C*-Algebras Banach *-Algebras and C*-Algebras Commutative C*-Algebras and Continuous Functional Calculus Algebraic and Order Theory of C*-Algebras Representations of Banach *-Algebras and C*-Algebras Universal and Enveloping C*-Algebras Exercises Nonlinear Functional Analysis Calculus on Banach Spaces Differentiable Functions Between Banach Spaces The Implicit and Inverse Function Theorems Integration in Banach Spaces Calculus of Variations The Isoperimetric Problem Measure and Topology; Generalized Functions and Distributions Measures on Locally Compact Spaces Regular Measures The Riesz Representation Theorem Distributions Generalized Functions on R Order of Generalized Functions and a Sequential Approach Distributions on Rn Tempered Distributions Sobolev Spaces Differential Equations and Distributions Ordinary Differential Equations and Generalized Functions Partial Differential Equations and Generalized Functions Topological Groups and Abstract Harmonic Analysis Topological Groups Groups and Homomorphisms Topological Groups Uniformities on Topological Groups Locally Compact Topological Groups Polish Groups and Baire Groups Open Mappings and Automatic Continuity Exercises Haar Measure Definition and Existence Haar Integrals Invariant Measures for Group Actions Properties of Haar Measure The Modular Function Measures on Homogeneous Spaces Extending Haar Measure Measurable Groups Exercises Group Algebras and Unitary Representations The Group Algebra The L1 Algebra and the C*-Algebra of a Locally Compact Group Lie Groups The Classical Groups Lie Groups and Lie Algebras Locally Euclidean Groups Locally Compact Abelian Groups and Pontryagin Duality The Dual Group Exercises Addenda Mathematical Names The Greek Alphabet Origin of Terms and Notation Miscellany Age and Mathematics The Mathematical Organ Mathematics and Mental Illness Literature Review Bibliography Index