Authors: Dean Corbae, Max Stinchcombe and Juraj Zeman
Type: e-book (final manuscript)
Level: Advanced Undergraduate (Math), MSc(Math. Fin), Ph.D.(Econ, Fin)
A manuscript (final, 2008) by Dean Corbae, Max Stinchcombe, and Juraj Zeman. After chapter 1 which briefly deals with the concept of logic, the authors cover set theory in chapter 2. In chapter 3 introduces the "Space of Real Numbers". You can read about the basic properties of rationals, the concept of distance, Cauchy sequences, supremum and infimum and the commpleteness of the Real Numbers. You can also find applications to economics. Chapter 4 is devoted to Metric Spaces $R^l,$ $l=1,2,...$ The basic definitions of Metric Spaces are introduced. You can also read about normed vector spaces, compacteness, completeness, closure, convergence and separability. Continuous functions on $R^l$ and Lipschitz and uniform continuity are covered next. There are also some applications to economic concepts. Convex analysis in $R^l$ is the topic of chapter 5 where you can read about convexity, the dual space of $R^l$, concave and convex functions, and the Hahn-Banach theorem. There are also many related to economics and optimization concepts such as the Kuhn-Tucker Theorem, Lagrange multipliers and fixed point theorems. Metric spaces is the subject of chapter 6. In chapter 7 the authors cover measure spaces and probability. There you can find the necessary background if you want to study stochastic calculus and option pricing. Measurable sets, probabilities, random variables, limit theorems, and convergence are just a small subset of the content. Chapter 8 is a little more technical and covers the $L^p(\Omega,\mathcal{F},P)$ and $l^p$ spaces, $p\in[1,\infty]$. There are applications to game theory and optimization. Chapters 9, 10 and 11 cover more advanced and technical concepts that a phd student in mathematical economics may find useful.
Type: e-book (final manuscript)
Level: Advanced Undergraduate (Math), MSc(Math. Fin), Ph.D.(Econ, Fin)
A manuscript (final, 2008) by Dean Corbae, Max Stinchcombe, and Juraj Zeman. After chapter 1 which briefly deals with the concept of logic, the authors cover set theory in chapter 2. In chapter 3 introduces the "Space of Real Numbers". You can read about the basic properties of rationals, the concept of distance, Cauchy sequences, supremum and infimum and the commpleteness of the Real Numbers. You can also find applications to economics. Chapter 4 is devoted to Metric Spaces $R^l,$ $l=1,2,...$ The basic definitions of Metric Spaces are introduced. You can also read about normed vector spaces, compacteness, completeness, closure, convergence and separability. Continuous functions on $R^l$ and Lipschitz and uniform continuity are covered next. There are also some applications to economic concepts. Convex analysis in $R^l$ is the topic of chapter 5 where you can read about convexity, the dual space of $R^l$, concave and convex functions, and the Hahn-Banach theorem. There are also many related to economics and optimization concepts such as the Kuhn-Tucker Theorem, Lagrange multipliers and fixed point theorems. Metric spaces is the subject of chapter 6. In chapter 7 the authors cover measure spaces and probability. There you can find the necessary background if you want to study stochastic calculus and option pricing. Measurable sets, probabilities, random variables, limit theorems, and convergence are just a small subset of the content. Chapter 8 is a little more technical and covers the $L^p(\Omega,\mathcal{F},P)$ and $l^p$ spaces, $p\in[1,\infty]$. There are applications to game theory and optimization. Chapters 9, 10 and 11 cover more advanced and technical concepts that a phd student in mathematical economics may find useful.
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