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 of the three "commonly examined" metrics,  showed that the “geometry” of the open (and/or the closed) balls crucially depends on the metric. Their geometric relations were later on exemplified via the notion of metric equivalence and its balls' implications. We thus examined the antimonotonic relationship between balls of the same centered arising by pairs of dominant and dominating metrics. We have used the norions of cylinders in order to obtain geometric depictions of balls wrt the uniform metric in convenient domains.

We begun our study of metric properties with the finitary notion of boundedness. The balls can be readily used in order to define it as a natural extension of the notion of boundness on the real line (w.r.t. the usual metric). Specifically a subset of a metric space is bounded  iff it can be covered by an open (equivalently closed) ball. 

We have begun to show several general poperties of the notion and provide with examples: e.g.  open and closed balls are by construction bounded. The notion is hereditary. If a set is bounded then any subset is also bounded. The dual is evident. If a set is not bounded then any superset is also unbounded. Obviously the notion depends crucially on the metric. For example the discrete space is always bounded. This is in contrast with any Euclidean space (i.e. the  equipped with the usual metric) which is not bounded-the same is true when equipped with any of the metrics we have been discussing except for the discrete one. 

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 (or anywhere inside the space as long as the radius id appropriately modified). We have shown that any finite set is (universally) bounded. In function spaces we have examined the notion of uniform boundedness (function boundedness via a common bound) via a variational property and found it to be conveniently equivalent to boundedness w.r.t. the uniform metric.

We have provided with further examples: Euclidean spaces are not bounded-the same is true when equipped with any of the metrics we have been discussing except for the discrete one. It is easy to see that Hamming spaces are bounded due to finiteness.

Notes on the above can be found here,  here, here, and here. 

Whiteboards from previous year's lectures (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found  here, here.

Exercise: Show that a subset A of a metric space X is bounded, iff it is a bounded space when considered as metric-subspace of X.