Math topics

Chapter and section references are from Corbae, Stinchcombe and Juraj (2009), “An Introduction to Mathematical Analysis for Economic Theory and Econometrics.'' I only recommend you follow Corbae et al. if you are already comfortable with highly-technical math. Otherwise, use this list of topics as guidance while using one of the recommended sources on the Textbooks page and following the guidance on the Math preparation page.


Chapter 1 in Corbae, Stinchcombe and Juraj (2009)

Statements, Sets, Subsets, ImplicationSection 1.1XXX
Ands, Ors, NotsSection 1.2.aXXX
Implies, EquivalenceSection 1.2.bXXX
Vacuous StatementsSection 1.2.cXXX
IndicatorsSection 1.2.dXX
Logical QuantifiersSection 1.4XX
Taxonomy of ProofsSection 1.5XXX

Set Theory

Chapter 2 in Stinchcombe and Juraj (2009)

Notation for sets2.2.2, top pg. 21XX
Useful theorems on sets2.2.4, 2.2.6X?
Cartesian Product2.3.1XXX
Relation2.3.4, 2.3.5X
Image2.3.16, 2.6.1, 2.6.4XX
Equivalence Class2.4.1, 2.4.5XXX
Inverse, Inverse Image2.6.7, 2.6.10, 2.6.13XXX
Level Sets of Functions2.6.12XXX
One-to-One / Injection2.6.15XX
Onto / Surjection / Bijection2.6.17XX
Composite Functions2.6.20, 2.6.23, 2.6.26XX

The Space of Real Numbers

Chapter 3 in Stinchcombe and Juraj (2009)

The `Why'Section 3.1 and 3.10X
Algebraic Properties of $\mathbb{Q}$3.2.3XX
Distance in $\mathbb{Q}$3.3.1, 3.3.2XX
Cauchy3.3.7, 3.4.8X
Bounded3.3.12, 3.3.13, 3.6.1XXX
Real Numbers3.3.19X
Algebraic Properties of $\mathbb{R}$3.3.23XX
Distance in $\mathbb{R}$3.4.1, 3.4.2, 3.4.3XX
Convergence3.4.9, 3.4.10, 3.4.15XXX
Completeness of $\mathbb{R}$3.4.16X
Supremum / Infimum3.6.2, 3.6.5XXX

The Finite-Dimensional Metric Space of Real Vectors

Chapter 4 in Stinchcombe and Juraj (2009)

Metric Space4.1.1X
Convergence, Limit4.1.4XXX
Complete4.1.6, 4.4.5XX
Open ball4.1.9X
Open4.1.10, 4.1.11XX
Open neighborhood4.1.12X
Open cover4.1.18X
Compact4.1.19, 4.7.15XX
Continuous4.1.22, 4.7.20, 4.85XXX
Vector Space4.3.1X?X
Normed Vector Space4.3.7X?
Inner / Dot Product4.3.9XXX
Cauchy-Schwartz Inequality4.3.10X
\textit{p}-NormsSection 4.3.cX?X
Characterizing Closed SetsSection 4.5.aXX
Closure of a Set4.5.4X
Boundary of a Set4.5.5XX
Accumulation / Cluster / Limit Point4.5.7XX
Closure and Completeness4.5.12, 4.5.13XX
Applications of CompactnessSection 4.7.fXXX
Basic Existence Result4.8.11, 4.8.16XX
Upper Hemicontinuity4.10.20XX
Theorem of the Maximum4.10.22, 6.1.31XX
Upper Semicontinuity4.10.29XX?
Connected4.1.12, 4.12.3, 4.12.4XXX
Interval4.12.1, 4.12.2XXX
Intermediate Value Theorem4.12.5XXX

Finite-Dimensional Convex Analysis

Chapter 5 in Stinchcombe and Juraj (2009)

Convex Combination5.1.2XXX
Convex Preferences and TechnologiesSection 5.1.cX
Returns to Scale5.1.13X
Convex Hull5.4.6X
Upper Contour Set5.4.23XXX
Affine Combination5.6.16X
Interior5.5.1, 5.5.2X
Concave Function5.6.1, 5.6.2XXX
Affine Function5.6.6XX
Implicit Function TheoremSections 2.8.a and 5.9.bXX
Envelope TheoremSection 5.9.cXX

Sections 5.8 - 5.10 contain results on optimization. The facts and results that you need should already be familiar from math camp so I do not list them separately.