Mathematical Economics

(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 with a motivation of further examination of such relations.

We begun the examination of the important example of the space of bounded real functions on a non-empty domain endowed wi

After some brief discussion of the course's scope and aims, and using the overview of the familiar case of the real numbers, 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 o

You can find the lectures' whiteboards here. Notes for the above can be found here and here.  

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

We continued with issues of (sequential) convergence in metric spaces. We examined the notion of continuity of functions between metric spaces. We then focused on our first major application: approximation of optimization problems.

You can find the lectures' whiteboards here. Notes for the above can be found here and 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

Strenghtening the boundedness characterization we have introduced the notion of total boundness. We have moved on to our examination of the latter. Given the analytical complexity of the verification of total boundness, we need the introduction of the notions of covering numbers and metric entropy in order to construct examples and further properties.

The lectures whiteboards can be found here. Notes on the above can be found here, and here.