A refresher on math Flashcards
Vector space definition
Distance
Normed vector space
Limit of a sequence
Limit point
Limit points of a convergent sequence
A convergent sequence in R_n can have at most one limit point
Theorem 3 Convergence and boundedness
Every convergent subsequence in R_n is bounded
Definition Cauchy sequence
Theorem 5 convergence of cauchy sequences
Every cauchy sequence in a metric space
- The sequence is bounded
- The sequence has at most one limit point
Def complete space
Every Cauchy sequence is convergent (to a point within the metric space)
Continuity of a mapping at a point
Pointwise convergence
Notice pointwise convergence doesn’t preserve continuity
Uniform convergence
Theorem 5 uniform convergence
Uniform convergence theorem
C(X)
Def contraction mapping
Banach fixed point theorem
Corollary Banach
Blackwell’s sufficiency conditions
Definition correspondence
Definitions of correspondence:
- compact valued
- closed valued
- convex valued
Their images are respectively:
- compact sets
- closed sets
- convex sets
Graph of a correspondence
Analogously, we have:
- Closed-graph correspondences
- Convex-graph correspondences
Lower hemi-continuous
lhc fails if there are discontinuities that blow up the “upper border” of the graph
Upper hemi-continuous sequence
Continuous correspondence
The correspondence is continuous at x if it is both uhc and lhc
Theorem 11: theorem of maximum
Corollary 12
Dynamic programming: assumptions
Theorem 13 on bellman operator
Assumptions 3-4
Under assumptions 1-4, V is strictly increasing
Theorem 14: V strictly increasing
Under assumptions 1-4, V is strictly increasing
Assumptions 5-6
Theorem 15: concavity of V
Theorem 16: uniform convergence of the policy function
Assumption 7
F is continuously differentiable in the interior of A
Theorem 17 (Benveniste and Scheinkman)
Theorem 18 (envelope)
