Course : Mathematical Economics

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 the uniform metric and noted that it contains several subexamples, e.g. the spaces of real n-vectors equipped with the max-metric.

Studying the example of the n-dimensional Hamming space, we introduced the useful notion of a metric sub-space. 

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.

Notes on the previous can be found  here and here.  The whiteboards from analogous previous year's lectures (please keep in mind that those arenot necessarily identical to the current lectures but they contain some common elements) can be found here and here.

Exercise

Show that  for some appropriate .