Mathematical Economics

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 agree on what parts of the metric space are considered bounded.

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. Hamming spaces are bounded due to finiteness. For function spaces with the uniform metric we introduced the notion of uniform boundedness and have proven that this variational property is equivalent to boundedness w.r.t. the particular metric. 

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

The whiteboards from past 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.