Mathematical Economics

We have examined further details on boundness including some limiting variance and a characterization that is useful for the introduction of total boundness. We have moved on to our examination of the latter. Given the analytical complexity of the verification of total boundness, we have briefly been occupied with the notions of covering numbers and metric entropy.

The lectures whiteboards can be found here and here. Notes on the above can be found here (for a generalization see here), here, her

We have examined several examples of boundness in function spaces using uniform boundness, We have also examined the hereditarity of boundness between metrics that satisfy functional inequality relations.

The lectures whiteboards can be found here. Notes on the above can be found here and here (you can also look for a generalization here).

Total boundness is a refinement of boundness. The latter is equivalent to the existence of some ε>0 for which there exists a finite cover of open (or equivalently closed) balls for the set at hand. Total boundness strengthens this by requiring that for any ε>0 there exists an analogous finite cover. By definition, for any ε>0, the analogous cover would be essentially determined by the finite set of ball centers, that a. can depend on ε, and b. need not lie inside the set at hand (but we can forc

You can find here and here exercises corresponding to the notions that were examined in the first three lectures.

We have completed our basic vocabulary regarding balls in metric spaces by showing that those can be used in order to seperate points. This property (which does not generally hold for pseudo-metrics) is essential, since it among others provides with the uniqueness of limits in metric spaces.

We begun our study of metric properties with the finitary notion of boundness. The balls can be readily used in order to define it as a natural extension of the notion of boundness on the real line (w.r.t. t

We completed our general definitions with the notion of the metric subspace. This as well as  the notion of product metric spaces (with a finite number of factors) can be viewed as ways to construct further metric spaces from given one(s), with the resulting metrics carrying relevant information on the given one(s).

We begun studying properties of metric spaces via the definition of the open and the closed balls that the metric defines. We have shown that these cannot in any case be empty, and o

After some brief discussion of the course's scope and aims, we begun with the definition of a distance function (metric) w.r.t. a non empty set of reference as a real function defined on the product of this set with itself that satisfies positivity, separation, symmetry and triangle inequality. The example of the discrete metric showed that any such set bears at least one such function, and further examples implied that more than one metrics can exist, possibly inside structured families, some o

In the final lecture we initially revisited the issue of the uniform convergence of the self-composition of a contraction inside a complete and bounded metric space in order i) to provide with a more efficient description compared to the one provided in the previous lecture, and ii) to remark that the BFPT essentially implied the pointwise convergence (why?), while the further attribute of boundedness for the space strengthened the latter to uniform convergence. You can find notes for the above

In our final TA session we exerted effort into understanding how solutions to Bellman equations relate to solutions of a standard class of Stationary Dynamic Programming Problems. For this purpose, we showed two lemmas related to dynamic programming and used them along with what we have seen in class about bellman equations in continuous and bounded functional space. We saw under what conditions the solution in question exists and is unique. We also solved an example dynamic programming problem.

We have been occupied with issues involving the uniform convergence of uniformly Lipschitz functions, and the uniform convergence of the self-composition of a contraction inside a bounded and complete metric space (are those two issues somehow related?).

We have begun studying the framework for applications of the BFPT, derived the useful Blackwell's Lemma, and begun the application concerning the uniqueness of the solution to the Bellman equation.

You can find notes for the above here, here (th