Course : Mathematical Economics

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. 

We have examined further details on boundness including: several (counter-) examples of boundness in function spaces using uniform boundness, some limiting variance, a characterization of boundedness for metric subspaces, and an equivalent definition that is useful for the introduction of total boundness.

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

We continued our study of ball properties and have shown that the local information that they convey about their center, can be conveyed by a "countable description", as well as that there exists an antimonotone relation between the open (closed) balls of fixed center and radius w.r.t. metrics that obey functional inequalities confirming the aforementioned remark concerning . 

We begun our study of metric properties with the finitary notion of boundness. The balls can be readily used in order to

We examined the important example of the space of bounded real functions on a non-empty domain endowed with the uniform metric and showed that it contains several subexamples, e.g. the spaces of real n-vectors equipped with the max-metric.

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 m

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