Course : Mathematical Economics

We continued studying properties of metric induced balls. We have shown that these obey some monotonicity property. The examples examined including the ones of the real line endowed with the usual metric, the real line endowed with the "exponential metric", the real line endowed with the discrete one, showed that the “geometry” of the open (and/or the closed) balls crucially depends on the metric.

We have provided with the geometric realizations of open and closed balls in  endowed with any

Given relations we have established in our real vector spaces examples, we have defined the notions of dominance and equivalence between metrics definable on the same carrier set. Two such metrics can be considered equivalent-and we suspected that the notion is representing "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 th

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

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 metri

Notes for the above can be found here, here and here. The whiteboards from analogous Ac. year's 2020-21 lectures (please keep in mind that those are not necessarily identical to the current lectures but they contain som

We provided a brief reminder of  topological notions in metric spaces with emphasis on the notion of (sequential) convergence and function continuity. Specifying the above in functional speces equipped with the uniform metric we have focused on the notion of uniform functional convergence constrasting it to the (generically non metrizable) notion of pointwise convergence. We then focused on a major application: the approximation of optimization problems.

Notes for the above can be found here and

We have continued with our application of the notion of total boundedness in asymptotic analysis: the derivation of a Uniform Law of Large Numbers in a simple framework using a chaining argument that makes use of finite coverings. We have constructed examples of totally bounded sets via the notion of covering numbers.

Notes on the above can be found here. Notes on the particular ULLN application can be found here. 

The whiteboards from analogous lectures from the Av. Year 2020-21 (please keep in

We begun with an equivalent definition that is useful for the introduction of total boundness. Strenghtening thus, the boundedness characterization we have introduced the notion of total boundness. We have started  our examination of the latter. We have given properties, e.g. that the centers of the covering balls can be chosen inside the totally bounded set, as well as (trivial) examples-counterexamples (the analytical complexity of the notion clearly manifested itself on that counter examples

We continued our study of the notion of boundedness. We have proven that the center of the covering ball need not be an element of the subset at hand although we have shown that when such a ball exists the center can always be chosen to lie inside this subset. We have shown that any finite set is (universally) bounded. We have proven that if a space is bounded w.r.t. a dominant metric, then it is also bounded w.r.t. to the dominated one. An obvious corollary is that equivalent metrics totally ag