Real Analysis with Economic Applications

Preface xvii
Prerequisites xxvii
Basic Conventions xxix

Part I: SET THEORY 1

Chapter A: Preliminaries of Real Analysis 3
A.1 Elements of Set Theory 4
A.1.1 Sets 4
A.1.2 Relations 9
A.1.3 Equivalence Relations 11
A.1.4 Order Relations 14
A.1.5 Functions 20
A.1.6 Sequences, Vectors, and Matrices 27
A.1.7* A Glimpse of Advanced Set Theory: The Axiom of Choice 29

A.2 Real Numbers 33
A.2.1 Ordered Fields 33
A.2.2 Natural Numbers, Integers, and Rationals 37
A.2.3 Real Numbers 39
A.2.4 Intervals and R 44

A.3 Real Sequences 46
A.3.1 Convergent Sequences 46
A.3.2 Monotonic Sequences 50
A.3.3 Subsequential Limits 53
A.3.4 Infinite Series 56
A.3.5 Rearrangement of Infinite Series 59
A.3.6 Infinite Products 61

A.4 Real Functions 62
A.4.1 Basic Definitions 62
A.4.2 Limits, Continuity, and Differentiation 64
A.4.3 Riemann Integration 69
A.4.4 Exponential, Logarithmic, and Trigonometric Functions 74
A.4.5 Concave and Convex Functions 77
A.4.6 Quasiconcave and Quasiconvex Functions 80

Chapter B: Countability 82
B.1 Countable and Uncountable Sets 82
B.2 Losets and Q 90

B.3 Some More Advanced Set Theory 93
B.3.1 The Cardinality Ordering 93
B.3.2* The Well-Ordering Principle 98

B.4 Application: Ordinal Utility Theory 99
B.4.1 Preference Relations 100
B.4.2 Utility Representation of Complete Preference Relations 102
B.4.3* Utility Representation of Incomplete Preference Relations 107

Part II: ANALYSIS ON METRIC SPACES 115

Chapter C: Metric Spaces 117
C.1 Basic Notions 118
C.1.1 Metric Spaces: Definition and Examples 119
C.1.2 Open and Closed Sets 127
C.1.3 Convergent Sequences 132
C.1.4 Sequential Characterization of Closed Sets 134
C.1.5 Equivalence of Metrics 136

C.2 Connectedness and Separability 138
C.2.1 Connected Metric Spaces 138
C.2.2 Separable Metric Spaces 140
C.2.3 Applications to Utility Theory 145

C.3 Compactness 147
C.3.1 Basic Definitions and the Heine-Borel Theorem 148
C.3.2 Compactness as a Finite Structure 151
C.3.3 Closed and Bounded Sets 154

C.4 Sequential Compactness 157
C.5 Completeness 161
C.5.1 Cauchy Sequences 161
C.5.2 Complete Metric Spaces: Definition and Examples 163
C.5.3 Completeness versus Closedness 167
C.5.4 Completeness versus Compactness 171

C.6 Fixed Point Theory I 172
C.6.1 Contractions 172
C.6.2 The Banach Fixed Point Theorem 175
C.6.3* Generalizations of the Banach Fixed Point Theorem 179

C.7 Applications to Functional Equations 183
C.7.1 Solutions of Functional Equations 183
C.7.2 Picard's Existence Theorems 187

C.8 Products of Metric Spaces 192
C.8.1 Finite Products 192
C.8.2 Countably Infinite Products 193

Chapter D: Continuity I 200
D.1 Continuity of Functions 201
D.1.1 Definitions and Examples 201
D.1.2 Uniform Continuity 208
D.1.3 Other Continuity Concepts 210
D.1.4* Remarks on the Differentiability of Real Functions 212
D.1.5 A Fundamental Characterization of Continuity 213
D.1.6 Homeomorphisms 216

D.2 Continuity and Connectedness 218
D.3 Continuity and Compactness 222
D.3.1 Continuous Image of a Compact Set 222
D.3.2 The Local-to-Global Method 223
D.3.3 Weierstrass' Theorem 225
D.4 Semicontinuity 229

D.5 Applications 237
D.5.1* Caristi's Fixed Point Theorem 238
D.5.2 Continuous Representation of a Preference Relation 239
D.5.3* Cauchy's Functional Equations: Additivity on Rn 242
D.5.4* Representation of Additive Preferences 247

D.6 CB(T) and Uniform Convergence 249
D.6.1 The Basic Metric Structure of CB(T) 249
D.6.2 Uniform Convergence 250
D.6.3* The Stone-Weierstrass Theorem and Separability of C(T) 257
D.6.4* The Arzelà-Ascoli Theorem 262
D.7* Extension of Continuous Functions 266

D.8 Fixed Point Theory II 272
D.8.1 The Fixed Point Property 273
D.8.2 Retracts 274
D.8.3 The Brouwer Fixed Point Theorem 277
D.8.4 Applications 280

Chapter E: Continuity II 283
E.1 Correspondences 284
E.2 Continuity of Correspondences 287
E.2.1 Upper Hemicontinuity 287
E.2.2 The Closed Graph Property 294
E.2.3 Lower Hemicontinuity 297
E.2.4 Continuous Correspondences 300
E.2.5* The Hausdorff Metric and Continuity 302
E.3 The Maximum Theorem 306

E.4 Application: Stationary Dynamic Programming 311
E.4.1 The Standard Dynamic Programming Problem 312
E.4.2 The Principle of Optimality 315
E.4.3 Existence and Uniqueness of an Optimal Solution 320
E.4.4 Application: The Optimal Growth Model 324

E.5 Fixed Point Theory III 330
E.5.1 Kakutani's Fixed Point Theorem 331
E.5.2* Michael's Selection Theorem 333
E.5.3* Proof of Kakutani's Fixed Point Theorem 339
E.5.4* Contractive Correspondences 341

E.6 Application: The Nash Equilibrium 343
E.6.1 Strategic Games 343
E.6.2 The Nash Equilibrium 346
E.6.3* Remarks on the Equilibria of Discontinuous Games 351

Part III: ANALYSIS ON LINEAR SPACES 355

Chapter F: Linear Spaces 357
F.1 Linear Spaces 358
F.1.1 Abelian Groups 358
F.1.2 Linear Spaces: Definition and Examples 360
F.1.3 Linear Subspaces, Affine Manifolds, and Hyperplanes 364
F.1.4 Span and Affine Hull of a Set 368
F.1.5 Linear and Affine Independence 370
F.1.6 Bases and Dimension 375

F.2 Linear Operators and Functionals 382
F.2.1 Definitions and Examples 382
F.2.2 Linear and Affine Functions 386
F.2.3 Linear Isomorphisms 389
F.2.4 Hyperplanes, Revisited 392

F.3 Application: Expected Utility Theory 395
F.3.1 The Expected Utility Theorem 395
F.3.2 Utility Theory under Uncertainty 403

F.4* Application: Capacities and the Shapley Value 409
F.4.1 Capacities and Coalitional Games 410
F.4.2 The Linear Space of Capacities 412
F.4.3 The Shapley Value 415

Chapter G: Convexity 422
G.1 Convex Sets 423
G.1.1 Basic Definitions and Examples 423
G.1.2 Convex Cones 428
G.1.3 Ordered Linear Spaces 432
G.1.4 Algebraic and Relative Interior of a Set 436
G.1.5 Algebraic Closure of a Set 447
G.1.6 Finitely Generated Cones 450

G.2 Separation and Extension in Linear Spaces 454
G.2.1 Extension of Linear Functionals 455
G.2.2 Extension of Positive Linear Functionals 460
G.2.3 Separation of Convex Sets by Hyperplanes 462
G.2.4 The External Characterization of Algebraically Closed and Convex Sets 471
G.2.5 Supporting Hyperplanes 473
G.2.6* Superlinear Maps 476

G.3 Reflections on Rn 480
G.3.1 Separation in Rn 480
G.3.2 Support in Rn 486
G.3.3 The Cauchy-Schwarz Inequality 488
G.3.4 Best Approximation from a Convex Set in Rn 489
G.3.5 Orthogonal Complements 492
G.3.6 Extension of Positive Linear Functionals, Revisited 496

Chapter H: Economic Applications 498
H.1 Applications to Expected Utility Theory 499
H.1.1 The Expected Multi-Utility Theorem 499
H.1.2* Knightian Uncertainty 505
H.1.3* The Gilboa-Schmeidler Multi-Prior Model 509

H.2 Applications to Welfare Economics 521
H.2.1 The Second Fundamental Theorem of Welfare Economics 521
H.2.2 Characterization of Pareto Optima 525
H.2.3* Harsanyi's Utilitarianism Theorem 526

H.3 An Application to Information Theory 528
H.4 Applications to Financial Economics 535
H.4.1 Viability and Arbitrage-Free Price Functionals 535
H.4.2 The No-Arbitrage Theorem 539
H.5 Applications to Cooperative Games 542
H.5.1 The Nash Bargaining Solution 542
H.5.2* Coalitional Games without Side Payments 546

Part IV: ANALYSIS ON METRIC/NORMED LINEAR SPACES 551

Chapter I: Metric Linear Spaces 553
I.1 Metric Linear Spaces 554
I.2 Continuous Linear Operators and Functionals 561
I.2.1 Examples of (Dis-)Continuous Linear Operators 561
I.2.2 Continuity of Positive Linear Functionals 567
I.2.3 Closed versus Dense Hyperplanes 569
I.2.4 Digression: On the Continuity of Concave Functions 573
I.3 Finite-Dimensional Metric Linear Spaces 577
I.4* Compact Sets in Metric Linear Spaces 582
I.5 Convex Analysis in Metric Linear Spaces 587
I.5.1 Closure and Interior of a Convex Set 587
I.5.2 Interior versus Algebraic Interior of a Convex Set 590
I.5.3 Extension of Positive Linear Functionals, Revisited 594
I.5.4 Separation by Closed Hyperplanes 594
I.5.5* Interior versus Algebraic Interior of a Closed and Convex Set 597

Chapter J: Normed Linear Spaces 601
J.1 Normed Linear Spaces 602
J.1.1 A Geometric Motivation 602
J.1.2 Normed Linear Spaces 605
J.1.3 Examples of Normed Linear Spaces 607
J.1.4 Metric versus Normed Linear Spaces 611
J.1.5 Digression: The Lipschitz Continuity of Concave Maps 614

J.2 Banach Spaces 616
J.2.1 Definition and Examples 616
J.2.2 Infinite Series in Banach Spaces 618
J.2.3* On the "Size" of Banach Spaces 620

J.3 Fixed Point Theory IV 623
J.3.1 The Glicksberg-Fan Fixed Point Theorem 623
J.3.2 Application: Existence of the Nash Equilibrium, Revisited 625
J.3.3* The Schauder Fixed Point Theorems 626
J.3.4* Some Consequences of Schauder's Theorems 630
J.3.5* Applications to Functional Equations 634

J.4 Bounded Linear Operators and Functionals 638
J.4.1 Definitions and Examples 638
J.4.2 Linear Homeomorphisms, Revisited 642
J.4.3 The Operator Norm 644
J.4.4 Dual Spaces 648
J.4.5* Discontinuous Linear Functionals, Revisited 649

J.5 Convex Analysis in Normed Linear Spaces 650
J.5.1 Separation by Closed Hyperplanes, Revisited 650
J.5.2* Best Approximation from a Convex Set 652
J.5.3 Extreme Points 654

J.6 Extension in Normed Linear Spaces 661
J.6.1 Extension of Continuous Linear Functionals 661
J.6.2* Infinite-Dimensional Normed Linear Spaces 663
J.7* The Uniform Boundedness Principle 665

Chapter K: Differential Calculus 670
K.1 Fréchet Differentiation 671
K.1.1 Limits of Functions and Tangency 671
K.1.2 What Is a Derivative? 672
K.1.3 The Fréchet Derivative 675
K.1.4 Examples 679
K.1.5 Rules of Differentiation 686
K.1.6 The Second Fréchet Derivative of a Real Function 690
K.1.7 Differentiation on Relatively Open Sets 694

K.2 Generalizations of the Mean Value Theorem 698
K.2.1 The Generalized Mean Value Theorem 698
K.2.2* The Mean Value Inequality 701

K.3 Fréchet Differentiation and Concave Maps 704
K.3.1 Remarks on the Differentiability of Concave Maps 704
K.3.2 Fréchet Differentiable Concave Maps 706

K.4 Optimization 712
K.4.1 Local Extrema of Real Maps 712
K.4.2 Optimization of Concave Maps 716

K.5 Calculus of Variations 718
K.5.1 Finite-Horizon Variational Problems 718
K.5.2 The Euler-Lagrange Equation 721
K.5.3* More on the Sufficiency of the Euler-Lagrange Equation 733
K.5.4 Infinite-Horizon Variational Problems 736
K.5.5 Application: The Optimal Investment Problem 738
K.5.6 Application: The Optimal Growth Problem 740
K.5.7* Application: The Poincaré-Wirtinger Inequality 743

Hints for Selected Exercises 747
References 777
Glossary of Selected Symbols 789
Index 793