Real Analysis with Economic Applications 1st edition by Efe A Ok – Ebook PDF Instant Download/Delivery: 0691117683 , 9780691117683
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ISBN 10: 0691117683
ISBN 13: 9780691117683
Author: Efe A Ok
There are many mathematics textbooks on real analysis, but they focus on topics not readily helpful for studying economic theory or they are inaccessible to most graduate students of economics. Real Analysis with Economic Applications aims to fill this gap by providing an ideal textbook and reference on real analysis tailored specifically to the concerns of such students. The emphasis throughout is on topics directly relevant to economic theory. In addition to addressing the usual topics of real analysis, this book discusses the elements of order theory, convex analysis, optimization, correspondences, linear and nonlinear functional analysis, fixed-point theory, dynamic programming, and calculus of variations. Efe Ok complements the mathematical development with applications that provide concise introductions to various topics from economic theory, including individual decision theory and games, welfare economics, information theory, general equilibrium and finance, and intertemporal economics. Moreover, apart from direct applications to economic theory, his book includes numerous fixed point theorems and applications to functional equations and optimization theory. The book is rigorous, but accessible to those who are relatively new to the ways of real analysis. The formal exposition is accompanied by discussions that describe the basic ideas in relatively heuristic terms, and by more than 1,000 exercises of varying difficulty. This book will be an indispensable resource in courses on mathematics for economists and as a reference for graduate students working on economic theory.
Real Analysis with Economic Applications 1st Table of contents:
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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