Mathematical Economics

After generalizing our basic definitions with the notions of a pseudo-metric and the subsequent notion of a psedo-metric space, as well as with the notion of a metric-subspace, we begun the examination of examples of metrics on spaces comprised of real (finite dimensional) vectors, including the Hamming distance. Our examples showed that several distinct metric spaces (over the same carrier) can become identical when restricted on particular sub-spaces.

Notes on the previous here, and a countere

After a presentation 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) over 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 metric

Notes for the above can be found here and here. 

The whiteboards from analogous previous year's lectures (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) 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 a major application: the approximation of optimization problems.

Notes for the above can be found here and here. The whiteboards from analogous previous year's lectures (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found here.  

We have begun our examination of topological notions in metric spaces with the notion of (sequential) convergence after establishing the Hausdorff and the first countability properties of metric spaces.

Notes on the above can be found here. The whiteboards from analogous previous year's lectures (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found here.

We have examined further details on total boundness among others involving, the comparison of metrics, as well as a 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 derived an application of the notion of total boundedness in asymptotic analysis, by deriving a Uniform Law of Large Numbers in a simple framework.

Notes on the

We show 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 boundedness. 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. the usual metric). Specifically a subset of a metric space is bounded  iff it can

We continued examining subexamples inside the important example of the space of bounded real functions on a non-empty domain endowed with the uniform metric.

We completed our general definitions with the notion of the metric subspace.

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 obey some monotonicity property. The examples of the real line endowed with the usua