Course : Mathematical Economics

We went through with the examination of several examples of such spaces, including the Hamming distance, as well as distances defined on sets of real finite dimensional vectors.

(Some of) The examples have shown that it is possible that different metrics on the same carrier set can obey relations, e.g. in the form of functional inequalities. We suspected that such relations might imply analogous ones between the relevant properties that each metric endows the space with, and that provides as wit

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

We continued with issues of (sequential) convergence in metric spaces.

You can find the lecture's whiteboards here. Notes for the above can be found here.  

We have examined further details on total boundness among others involving our discriptive examination of the notion of covering numbers. We have pointed out to the usefulness of the above in applications involving issues of convergence in metric spaces, or properties of stochastic properties. We have begun our examination of topological notions in metric spaces with the notion of (sequential) convergence.

The lecture's whiteboards can be found here. Notes on the above can be found here, and her

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