Course : Mathematical Economics

We have provided with the geometric realizations of open and closed balls in  endowed with any of the three "commonly examined" metrics,  showed that the “geometry” of the open (and/or the closed) balls crucially depends on the metric. Their geometric relations were later on exemplified via the notion of metric equivalence and its balls' implications. We thus examined the antimonotonic relationship between balls of the same centered arising by pairs of dominant and dominating metrics.

We begun o

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 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 usual metric, the real line endowed with the "exponential metric", the real line

We have investigated the further relation notion between metrics on the same carrier set; two suchlike metrics are considered equivalent whenever the first dominates the second and vice versa. Hence two such metrics can be considered equivalent-and suspected of introducing "similar properties" on the same carrier-without being necessarily equal as functions.

We examined the important-to the upcoming lectures-example of the space of bounded real functions on a non-empty domain endowed with the un

We continued the investigation of examples involving metrics on sets of finite dimensional real 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 with a motivation of further examination of such relations.

Notes on the pre

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.