Real Analysis With Economic Applications by Ok, Efe A. -- Read -- Imperial Library of Trantor

Index

Cover Page Title Page Copyright Page Dedication Page Contents Preface Prerequisites Basic Conventions Part I - Set Theory

Chapter A - Preliminaries of Real Analysis

A.1 - Elements of Set Theory

A.1.1 - Sets A.1.2 - Relations A.1.3 - Equivalence Relations A.1.4 - Order Relations A.1.5 - Functions A.1.6 - Sequences, Vectors, and Matrices A.1.7* - A Glimpse of Advanced Set Theory: The Axiom of Choice

A.2 - Real Numbers

A.2.1 - Ordered Fields A.2.2 - Natural Numbers, Integers, and Rationals A.2.3 - Real Numbers A.2.4 - Intervals and R

A.3 - Real Sequences

A.3.1 - Convergent Sequences A.3.2 - Monotonic Sequences A.3.3 - Subsequential Limits A.3.4 - Infinite Series A.3.5 - Rearrangement of Infinite Series A.3.6 - Infinite Products

A.4 - Real Functions

A.4.1 - Basic Definitions A.4.2 - Limits, Continuity, and Differentiation A.4.3 - Riemann Integration A.4.4 - Exponential, Logarithmic, and Trigonometric Functions A.4.5 - Concave and Convex Functions A.4.6 - Quasiconcave and Quasiconvex Functions

Chapter B - Countability

B.B.1 - Countable and Uncountable Sets B.B.2 - Losets and Q B.B.3 - Some More Advanced Set Theory

B.3.1 - The Cardinality Ordering B.3.2* - The Well-Ordering Principle

B.4 - Application: Ordinal Utility Theory

B.4.1 - Preference Relations B.4.2 - Utility Representation of Complete Preference Relations B.4.3* - Utility Representation of Incomplete Preference Relations

Part II - Analysis on Metric Spaces

Chapter C - Metric Spaces

C.1 - Basic Notions

C.1.1 - Metric Spaces: Definition and Examples C.1.2 - Open and Closed Sets C.1.3 - Convergent Sequences C.1.4 - Sequential Characterization of Closed Sets C.1.5 - Equivalence of Metrics

C.2 - Connectedness and Separability

C.2.1 - Connected Metric Spaces C.2.2 - Separable Metric Spaces C.2.3 - Applications to Utility Theory

C.3 - Compactness

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

C.4 - Sequential Compactness C.5 - Completeness

C.5.1 - Cauchy Sequences C.5.2 - Complete Metric Spaces: Definition and Examples C.5.3 - Completeness versus Closedness C.5.4 - Completeness versus Compactness

C.6 - Fixed Point Theory I

C.6.1 - Contractions C.6.2 - The Banach Fixed Point Theorem C.6.3* - Generalizations of the Banach Fixed Point Theorem

C.7 - Applications to Functional Equations

C.7.1 - Solutions of Functional Equations C.7.2 - Picard's Existence Theorems

C.8 - Products of Metric Spaces

C.8.1 - Finite Products C.8.2 - Countably Infinite Products

Chapter D - Continuity I

D.1 - Continuity of Functions

D.1.1 - Definitions and Examples D.1.2 - Uniform Continuity D.1.3 - Other Continuity Concepts D.1.4* - Remarks on the Differentiability of Real Functions D.1.5 - A Fundamental Characterization of Continuity D.1.6 - Homeomorphisms

D.2 - Continuity and Connectedness D.3 - Continuity and Compactness

D.3.1 - Continuous Image of a Compact Set D.3.2 - The Local-to-Global Method D.3.3 - Weierstrass' Theorem

D.4 - Semicontinuity D.5 - Applications

D.5.1* - Caristi's Fixed Point Theorem D.5.2 - Continuous Representation of a Preference Relation D.5.3* - Cauchy's Functional Equations: Additivity on Rn D.5.4* - Representation of Additive Preferences

D.6 - CB(T) and Uniform Convergence

D.6.1 - The Basic Metric Structure of CB (T) D.6.2 - Uniform Convergence D.6.3* - The Stone-Weierstrass Theorem and Separability of C(T) D.6.4* - The Arzelà-Ascoli Theorem

D.7* - Extension of Continuous Functions D.8 - Fixed Point Theory II

D.8.1 - The Fixed Point Property D.8.2 - Retracts D.8.3 - The Brouwer Fixed Point Theorem D.8.4 - Applications

Chapter E - Continuity II

E.1 - Correspondences E.2 - Continuity of Correspondences

E.2.1 - Upper Hemicontinuity E.2.2 - The Closed Graph Property E.2.3 - Lower Hemicontinuity E.2.4 - Continuous Correspondences E.2.5* - The Hausdorff Metric and Continuity

E.3 - The Maximum Theorem E.4 - Application: Stationary Dynamic Programming

E.4.1 - The Standard Dynamic Programming Problem E.4.2 - The Principle of Optimality E.4.3 - Existence and Uniqueness of an Optimal Solution E.4.4 - Application: The Optimal Growth Model

E.5 - Fixed Point Theory III

E.5.1 - Kakutani's Fixed Point Theorem E.5.2* - Michael's Selection Theorem E.5.3* - Proof of Kakutani's Fixed Point Theorem E.5.4* - Contractive Correspondences

E.6 - Application: The Nash Equilibrium

E.6.1 - Strategic Games E.6.2 - The Nash Equilibrium E.6.3* - Remarks on the Equilibria of Discontinuous Games

Part III - Analysis on Linear Spaces

Chapter F - Linear Spaces

F.1 - Linear Spaces

F.1.1 - Abelian Groups F.1.2 - Linear Spaces: Definition and Examples F.1.3 - Linear Subspaces, Affine Manifolds, and Hyperplanes F.1.4 - Span and Affine Hull of a Set F.1.5 - Linear and Affine Independence F.1.6 - Bases and Dimension

F.2 - Linear Operators and Functionals

F.2.1 - Definitions and Examples F.2.2 - Linear and Affine Functions F.2.3 - Linear Isomorphisms F.2.4 - Hyperplanes, Revisited

F.3 - Application: Expected Utility Theory

F.3.1 - The Expected Utility Theorem F.3.2 - Utility Theory under Uncertainty

F.4* - Application: Capacities and the Shapley Value

F.4.1 - Capacities and Coalitional Games F.4.2 - The Linear Space of Capacities F.4.3 - The Shapley Value

Chapter G - Convexity

G.1 - Convex Sets

G.1.1 - Basic Definitions and Examples G.1.2 - Convex Cones G.1.3 - Ordered Linear Spaces G.1.4 - Algebraic and Relative Interior of a Set G.1.5 - Algebraic Closure of a Set G.1.6 - Finitely Generated Cones

G.2 - Separation and Extension in Linear Spaces

G.2.1 - Extension of Linear Functionals G.2.2 - Extension of Positive Linear Functionals G.2.3 - Separation of Convex Sets by Hyperplanes G.2.4 - The External Characterization of Algebraically Closed and Convex Sets G.2.5 - Supporting Hyperplanes G.2.6* - Superlinear Maps

G.3 - Reflections on Rn

G.3.1 - Separation in Rn G.3.2 - Support in Rn G.3.3 - The Cauchy-Schwarz Inequality G.3.4 - Best Approximation from a Convex Set in Rn G.3.5 - Orthogonal Complements G.3.6 - Extension of Positive Linear Functionals, Revisited

Chapter H - Economic Applications

H.1 - Applications to Expected Utility Theory

H.1.1 - The Expected Multi-Utility Theorem H.1.2* - Knightian Uncertainty H.1.3* - The Gilboa-Schmeidler Multi-Prior Model

H.2 - Applications to Welfare Economics

H.2.1 - The Second Fundamental Theorem of Welfare Economics H.2.2 - Characterization of Pareto Optima H.2.3* - Harsanyi's Utilitarianism Theorem

H.3 - An Application to Information Theory H.4 - Applications to Financial Economics

H.4.1 - Viability and Arbitrage-Free Price Functionals H.4.2 - The No-Arbitrage Theorem

H.5 - Applications to Cooperative Games

H.5.1 - The Nash Bargaining Solution H.5.2* - Coalitional Games without Side Payments

Part IV - Analysis on Metric/Normed Linear Spaces

Chapter I - Metric Linear Spaces

I.1 - Metric Linear Spaces I.2 - Continuous Linear Operators and Functionals

I.2.1 - Examples of (Dis-)Continuous Linear Operators I.2.2 - Continuity of Positive Linear Functionals I.2.3 - Closed versus Dense Hyperplanes I.2.4 - Digression: On the Continuity of Concave Functions

I.3 - Finite-Dimensional Metric Linear Spaces I.4* - Compact Sets in Metric Linear Spaces I.5 - Convex Analysis in Metric Linear Spaces

I.5.1 - Closure and Interior of a Convex Set I.5.2 - Interior Versus Algebraic Interior of a Convex Set I.5.3 - Extension of Positive Linear Functionals, Revisited I.5.4 - Separation by Closed Hyperplanes I.5.5* - Interior versus Algebraic Interior of a Closed and Convex Set

Chapter J - Normed Linear Spaces

J.1 - Normed Linear Spaces

J.1.1 - A Geometric Motivation J.1.2 - Normed Linear Spaces J.1.3 - Examples of Normed Linear Spaces J.1.4 - Metric versus Normed Linear Spaces J.1.5 - Digression: The Lipschitz Continuity of Concave Maps

J.2 - Banach Spaces

J.2.1 - Definition and Examples J.2.2 - Infinite Series in Banach Spaces J.2.3* - On the “Size” of Banach Spaces

J.3 - Fixed Point Theory IV

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

J.4 - Bounded Linear Operators and Functionals

J.4.1 - Definitions and Examples J.4.2 - Linear Homeomorphisms, Revisited J.4.3 - The Operator Norm J.4.4 - Dual Spaces J.4.5* - Discontinuous Linear Functionals, Revisited

J.5 - Convex Analysis in Normed Linear Spaces

J.5.1 - Separation by Closed Hyperplanes, Revisited J.5.2* - Best Approximation from a Convex Set J.5.3 - Extreme Points

J.6 - Extension in Normed Linear Spaces

J.6.1 - Extension of Continuous Linear Functionals J.6.2* - Infinite-Dimensional Normed Linear Spaces

J.7* - The Uniform Boundedness Principle

Chapter K - Differential Calculus

K.1 - Fréchet Differentiation

K.1.1 - Limits of Functions and Tangency K.1.2 - What Is a Derivative? K.1.3 - The Fréchet Derivative K.1.4 - Examples K.1.5 - Rules of Differentiation K.1.6 - The Second Fréchet Derivative of a Real Function K.1.7 - Differentiation on Relatively Open Sets

K.2 - Generalizations of the Mean Value Theorem

K.2.1 - The Generalized Mean Value Theorem K.2.2* - The Mean Value Inequality

K.3 - Fréchet Differentiation and Concave Maps

K.3.1 - Remarks on the Differentiability of Concave Maps K.3.2 - Fréchet Differentiable Concave Maps

K.4 - Optimization

K.4.1 - Local Extrema of Real Maps K.4.2 - Optimization of Concave Maps

K.5 - Calculus of Variations

K.5.1 - Finite-Horizon Variational Problems K.5.2 - The Euler-Lagrange Equation K.5.3* - More on the Sufficiency of the Euler-Lagrange Equation K.5.4 - Infinite-Horizon Variational Problems K.5.5 - Application: The Optimal Investment Problem K.5.6 - Application: The Optimal Growth Problem K.5.7* - Application: The Poincaré-Wirtinger Inequality

Hints for Selected Exercises References Glossary of Selected Symbols Index

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