Real analysis
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.
Logic
Chapter 1 in Corbae, Stinchcombe and Juraj (2009)
| Topic | Reference | Background | Proofs | Direct/General |
|---|---|---|---|---|
| Statements, Sets, Subsets, Implication | Section 1.1 | X | X | X |
| Ands, Ors, Nots | Section 1.2.a | X | X | X |
| Implies, Equivalence | Section 1.2.b | X | X | X |
| Vacuous Statements | Section 1.2.c | X | X | X |
| Indicators | Section 1.2.d | X | X | |
| Logical Quantifiers | Section 1.4 | X | X | |
| Taxonomy of Proofs | Section 1.5 | X | X | X |
Set Theory
Chapter 2 in Stinchcombe and Juraj (2009)
| Topic | Reference | Background | Proofs | Direct/General |
|---|---|---|---|---|
| Notation for sets | 2.2.2, top pg. 21 | X | X | |
| Useful theorems on sets | 2.2.4, 2.2.6 | X? | ||
| Cartesian Product | 2.3.1 | X | X | X |
| Relation | 2.3.4, 2.3.5 | X | ||
| Function | 2.3.8 | X | X | X |
| Correspondence | 2.3.12 | X | X | X |
| Image | 2.3.16, 2.6.1, 2.6.4 | X | X | |
| Cardinality | 2.3.17 | X | X | X |
| Equivalence Class | 2.4.1, 2.4.5 | X | X | X |
| Partition | 2.4.9 | X | X? | |
| Inverse, Inverse Image | 2.6.7, 2.6.10, 2.6.13 | X | X | X |
| Level Sets of Functions | 2.6.12 | X | X | X |
| One-to-One / Injection | 2.6.15 | X | X | |
| Onto / Surjection / Bijection | 2.6.17 | X | X | |
| Composite Functions | 2.6.20, 2.6.23, 2.6.26 | X | X |
The Space of Real Numbers
Chapter 3 in Stinchcombe and Juraj (2009)
| Topic | Reference | Background | Proofs | Direct/General |
|---|---|---|---|---|
| The `Why' | Section 3.1 and 3.10 | X | ||
| Algebraic Properties of $\mathbb{Q}$ | 3.2.3 | X | X | |
| Distance in $\mathbb{Q}$ | 3.3.1, 3.3.2 | X | X | |
| Sequence | 3.3.3 | X | X | X |
| Subsequence | 3.3.5 | X | X | |
| Cauchy | 3.3.7, 3.4.8 | X | ||
| Bounded | 3.3.12, 3.3.13, 3.6.1 | X | X | X |
| Real Numbers | 3.3.19 | X | ||
| Algebraic Properties of $\mathbb{R}$ | 3.3.23 | X | X | |
| Distance in $\mathbb{R}$ | 3.4.1, 3.4.2, 3.4.3 | X | X | |
| Convergence | 3.4.9, 3.4.10, 3.4.15 | X | X | X |
| Completeness of $\mathbb{R}$ | 3.4.16 | X | ||
| Supremum / Infimum | 3.6.2, 3.6.5 | X | X | X |
The Finite-Dimensional Metric Space of Real Vectors
Chapter 4 in Stinchcombe and Juraj (2009)
| Topic | Reference | Background | Proofs | Direct/General |
|---|---|---|---|---|
| Metric Space | 4.1.1 | X | ||
| Convergence, Limit | 4.1.4 | X | X | X |
| Complete | 4.1.6, 4.4.5 | X | X | |
| Open ball | 4.1.9 | X | ||
| Open | 4.1.10, 4.1.11 | X | X | |
| Open neighborhood | 4.1.12 | X | ||
| Open cover | 4.1.18 | X | ||
| Compact | 4.1.19, 4.7.15 | X | X | |
| Connected | 4.1.21 | X | X | |
| Continuous | 4.1.22, 4.7.20, 4.85 | X | X | X |
| Vector Space | 4.3.1 | X? | X | |
| Normed Vector Space | 4.3.7 | X? | ||
| Inner / Dot Product | 4.3.9 | X | X | X |
| Cauchy-Schwartz Inequality | 4.3.10 | X | ||
| \textit{p}-Norms | Section 4.3.c | X? | X | |
| Characterizing Closed Sets | Section 4.5.a | X | X | |
| Closure of a Set | 4.5.4 | X | ||
| Boundary of a Set | 4.5.5 | X | X | |
| Accumulation / Cluster / Limit Point | 4.5.7 | X | X | |
| Closure and Completeness | 4.5.12, 4.5.13 | X | X | |
| Bounded | 4.7.8 | X | X | X |
| Applications of Compactness | Section 4.7.f | X | X | X |
| Basic Existence Result | 4.8.11, 4.8.16 | X | X | |
| Upper Hemicontinuity | 4.10.20 | X | X | |
| Theorem of the Maximum | 4.10.22, 6.1.31 | X | X | |
| Upper Semicontinuity | 4.10.29 | X | X? | |
| Connected | 4.1.12, 4.12.3, 4.12.4 | X | X | X |
| Interval | 4.12.1, 4.12.2 | X | X | X |
| Intermediate Value Theorem | 4.12.5 | X | X | X |
Finite-Dimensional Convex Analysis
Chapter 5 in Stinchcombe and Juraj (2009)
| Topic | Reference | Background | Proofs | Direct/General |
|---|---|---|---|---|
| Convex Combination | 5.1.2 | X | X | X |
| Convex Preferences and Technologies | Section 5.1.c | X | ||
| Returns to Scale | 5.1.13 | X | ||
| Convex Hull | 5.4.6 | X | ||
| Upper Contour Set | 5.4.23 | X | X | X |
| Affine Combination | 5.6.16 | X | ||
| Interior | 5.5.1, 5.5.2 | X | ||
| Concave Function | 5.6.1, 5.6.2 | X | X | X |
| Affine Function | 5.6.6 | X | X | |
| Quasi-Concave | 5.6.12 | X | X | X |
| Single-Peaked | 5.6.13 | X | ||
| Implicit Function Theorem | Sections 2.8.a and 5.9.b | X | X | |
| Envelope Theorem | Section 5.9.c | X | X |
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.