Mathematical Economics

We completed our general definitions with the notions of the metric subspaces and of the product metric spaces (with a finite number of factors). We can view the above as ways to construct further metric spaces from given one(s), with the resulting metrics carrying relevant information on the given one(s).

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, obey some monot

We have been occupied with further examples of metrics and subsequent metric spaces, and have shown that it is possible that different metrics on the same carrier set can obey relations. 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 examined the important example of the space of bounded real functions on a non-empty domain endow

After some brief discussion of the course's scope and aims, as well as of some prerequisite notions, 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, po

We have been further occupied with Brouwer's FPT providing an example that showed that it cannot generally hold in infinite dimensional spaces. We have subsequently examined an application of Brouwer's FPT application for the verification of the existence of Nash equilibria in a class of finite non-co-operative games, in a restricted setting that avoids the use of its generalization to correspendences, i.e. Kakutani's FPT. Our restrictive setting essentially ensures that the best response corres

We have been further occupied with the development of preparatory notions for the Brouwer's FPT. Those involved the topological notion of retraction. We have provided examples and counter examples. We have been examining the No Retraction Theorem-Borsuk's Lemma. We provided with a sketch of the proof using the concept of singular homology. Obviously this sketch of proof is out of the scope of the course. Using the latter theorem and the other preparatory notions we have proven the Brouwer Fixed

We have been occupied with a further application of the Banach FPT, namely the Picard-Lindelof Theorem. (Further corrected and supplemented) Notes of which you can find here. We begun our preparation for the establishment of the Brouwer FPT, introducing the notions of fixed point property and of that of topological homeomorphism. You can find (corrected) notes for the above here.  

We have been occupied with issues about metric fixed point theory involving the derivation of the Banach FPT, and applications w.r.t. the properties of the Bellman equation. You can find (corrected and supplemented) notes on the above here.

Given the examination of the non-topological notion of completeness of a metric space that was made available in the previous tutorial we were occupied with the issue of completeness (w.r.t. the relevant uniform metric) of a bounded function space when the range is a complete metric space. You can find notes for the aforementioned notions here (remember that the part of the proof of the latter result concerning the boundness of the limit of the relevant Cauchy sequence was left as an exercise).

Given the tutorial preparation of the notion of continuity of functions between metric spaces and a brief re-examination of this during the lecture, we have provided with a major application that establishes the continuity of the sup functional when properly restricted to possibly non-empty metric subspaces of the set of bounded real functions equiped with the uniform metric.

Using this and under further assumptions we were occupied with the issue of convergence of approximate maximizers under u

Given the general definition of a topological space we defined the concept of a metrizable topology (i.e. a topology generated by a metric). We derived an easy example concerning the indiscrete topology that showed the existence of topologies that are not metrizable. We were occupied with the issue of when two metrics define the same topology using the usual concepts of comparisons between metrics.

We begun the study of the concept of sequential convergence given the notion of a neighborhood sys