Course : Mathematical Economics

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

You can find here an example of the use of the metric entropy integral in the case of the Wiener process. We will work out this example when we examine the notion of continuity of functions between metric spaces. Furthermore, you can find here, a simple example, built upon an already examined one, that makes evident that non total boundness does not imply the existence of one element balls (as is typically the case in discrete spaces). Furthermore it implies that open balls can be non totally bo

We continued the examination of the notion of total boundness. We provided several remarks regarding hereditarity, etc, that implied that the notion coincides with that of boundness on Euclidean spaces. We briefly discussed the notion of metric entropy and described the Metric Entropy Integral Theorem of Dudley as an astonishing application of the notion of total boundness in the theory of Gaussian Processes. As previously you can find notes for total boundness here and for the aforementioned re

You can find here a generalization of the issue of metrics comparison w.r.t. boundness, here an example where the property is "somehow lost in the limit", and here a result on boundness obtained by some asymptotic comparison of balls between two metrics.

We have been occupied with further examples, and the issues of hereditarity on subsets of boundness, equivalence of metrics w.r.t. boundness, and the "invariance" of boundness w.r.t. finite products and the metrics we have defined on such products. You can find notes for the above here, here, and here.

We moved on to a refinement of the notion, obtaining the notion of total boundness, and we have initially provided with the non-equivalence of the two notions. You can find notes for the above her

You can find here the example of a metric space with a metric defined in such a way so that some of the open balls are non-convex (the carrier set is simply the reals-hence they have the algebraic structure for the notion of convexity to make sense), along with a generalization. This has emerged by a discussion with one of your collegues.

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 monotonicity property,  they can separate points in a metric space, while this does not generally hold for pseudo metrics, and the local information that they convey about their center, can be conveyed by a "countable description". The example of the real line endowed with the usual metric and the real line endo