Mathematical Economics

We have been initially occupied with the non-topological notion of Lipschitz continuity and the subsequent notion of a contractive self-map.

We have then been occupied with issues on metric fixed point theory involving the derivation of the Banach FPT, some straightforward corrolaries, and some initial applications.

You can find notes on the above here and here.

During the third TA session we were mostly preoccupied with proving a lemma concerning the completeness property of general bounded functional spaces (general as in not necessarily mapping on the real numbers), as it is inherited by its image space. You can find the above lemma and its proof here and at the end of these notes here.

We also quickly went through and commented on an economic application of the notions of pointwise versus uniform convergence of sequences of functions and what the

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

During the second TA session we were proeccopied with the following:

First, we revisited and corrected exercise 8 of the problem set 1 that was erroneously solved during the first TA session (find the solutions here).

Secondly, we proved two lemmas. The first concerns total boundness and finite product spaces, already seen in class. The proof is analogous to the proof of boundness and finite products. The second concerns the equivalence of characterising continuity using open balls or neigbo

We were occupied with the comparison of topologies generated by different by different metrics that obey functional inequalities.

We begun the study of the concept of sequential convergence given the notion of a neighborhood system of a point in a topological space and we have proven that in a metric space the notion can be equivalently described via the subsystem of open balls centered at the limit. We have proven that a sequence inside a metric space can have at most one limit, due to the prop

We concluded our examination of the notion of total boundness by considering issues involving metric comparisons and finite products. We provided several remarks regarding hereditarity, etc, that implied that the notion coincides with that of boundness on Euclidean spaces.

We begun the examination of topological notions on a metric space that arise by the presence of a metric. We initially defined the notion of open and closed subsets as duals, then the subsequent notion of a topology arising fr

On our first TA session we went through two sets of exercises on some elementary notions of metric spaces, also covered during the lectures. In particular we tried to firmly embed and understand the properties of metric functions and open (closed) balls in metric spaces.

You can find here and here solutions to all the exercises that we saw during the TA session. As mentioned in class, the rest of the exercises are either similar to those we solved or will not preoccupy as in general.

As was

We have been occupied with further examples, and the issues of 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 and here.

We have also been occupied with an example of a "sequence of metric spaces", where the property is "somehow lost in the limit", that can be found here. You can also find here a generalization of the issue of metrics comparison w.r.t. boundne

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. 

We begun our study of metric space properties with the finitary notion of boundness. The existence of the collection of the open balls in a metric space allows for the d

You can find here and here exercises corresponding to the notions that were examined in the first three lectures.