Mathematical Economics

We examined the important example of the space of bounded real functions on a non-empty domain endowed with the uniform metric and showed that it contains several subexamples, e.g. the spaces of real n-vectors equipped with the max-metric.

We completed our general definitions with the notion of the metric subspace. This as well as  the notion of product metric spaces (with a finite number of factors) can be viewed 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, 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 endowed with the discrete one, or of the  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. 

We have extended our basic vocabulary regarding balls in metric spaces by showing that those can be used in order to seperate points. This property (which does not generally hold for pseudo-metrics) is essential, since it among others provides with the uniqueness of limits in metric spaces.

Notes for the above can be found here and here.

The lecture's whiteboards can be found here.