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Unformatted text preview: . REAL ANALYSIS with ECONOMIC APPLICATIONS EFE A. OK New York University December, 2005 . ... mathematics is very much like poetry . .. what makes a good poem  a great poem  is that there is a large amount of thought expressed in very few words. In this sense formulas like e π i + 1 = 0 or ] ∞ −∞ e − x 2 dx = √ π are poems. Lipman Bers ii Contents Preface Chapter A Preliminaries of Real Analysis A.1 Elements of Set Theory 1 Sets 2 Relations 3 Equivalence Relations 4 Order Relations 5 Functions 6 Sequences, Vectors and Matrices 7 ∗ A Glimpse of Advanced Set Theory: The Axiom of Choice A.2 Real Numbers 1 Ordered Fields 2 Natural Numbers, Integers and Rationals 3 Real Numbers 4 Intervals and R A.3 Real Sequences 1 Convergent Sequences 2 Monotonic Sequences 3 Subsequential Limits 4 In f nite Series 5 Rearrangements of In f nite Series 6 In f nite Products A.4 Real Functions 1 Basic De f nitions 2 Limits, Continuity and Di ﬀ erentiation 3 Riemann Integration 4 Exponential, Logarithmic and Trigonometric Functions 5 Concave and Convex Functions 6 Quasiconcave and Quasiconvex Functions Chapter B Countability B.1 Countable and Uncountable Sets B.2 Losets and Q B.3 Some More Advanced Theory 1 The Cardinality Ordering 2 ∗ The Well Ordering Principle iii B.4 Application: Ordinal Utility Theory 1 Preference Relations 2 Utility Representation of Complete Preference Relations 3 ∗ Utility Representation of Incomplete Preference Relations Chapter C Metric Spaces C.1 Basic Notions 1 Metric Spaces: De f nitions and Examples 2 Open and Closed Sets 3 Convergent Sequences 4 Sequential Characterization of Closed Sets 5 Equivalence of Metrics C.2 Connectedness and Separability 1 Connected Metric Spaces 2 Separable Metric Spaces 3 Applications to Utility Theory C.3 Compactness 1 Basic De f nitions and the HeineBorel Theorem 2 Compactness as a Finite Structure 3 Closed and Bounded Sets C.4 Sequential Compactness C.5 Completeness 1 Cauchy Sequences 2 Complete Metric Spaces: De f nition and Examples 3 Completeness vs. Closedness 4 Completeness vs. Compactness C.6 Fixed Point Theory I 1 Contractions 2 The Banach Fixed Point Theorem 3 ∗ Generalizations of the Banach Fixed Point Theorem C.7 Applications to Functional Equations 1 Solutions of Functional Equations 2 Picard’s Existence Theorems C.8 Products of Metric Spaces 1 Finite Products 2 Countably In f nite Products Chapter D Continuity I D.1 Continuity of Functions iv 1 De f nitions and Examples 2 Uniform Continuity 3 Other Continuity Concepts 4 ∗ Remarks on the Di ﬀ erentiability of Real Functions 5 A Fundamental Characterization of Continuity 6 Homeomorphisms D.2 Continuity and Connectedness D.3 Continuity and Compactness 1 Continuous Image of a Compact Set 2 The LocaltoGlobal Method 3 Weierstrass’ Theorem D.4 Semicontinuity D.5 Applications 1 ∗ Caristi’s Fixed Point Theorem 2 Continuous Representation of a Preference Relation 3 ∗ Cauchy’s Functional Equations: Additivity on R n...
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 Spring '11
 zobuz
 Set Theory, Topology, The Land, Mathematical analysis, Metric space, Mathematical theorems

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