Mathematical Economics

We have been occupied with further examples of metrics and subsequent metric spaces, and have shown that it is possible that different metrics on the same carrier set can obey relations. 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.

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 as particular example the set of bounded real sequences with the relevant extension of the finite dimensional max-metric. We have also examined the subset of the latter consisting of the square summable real sequences and noticed that we can also define in this the obvious extension of the Euclidean metric. Furthermore, we have also examined the subset of the latter consisting of the absolutely summable real sequences and noticed that we can also define in this the obvious extension of the absolute metric. Notice that the last three examples essentially showed us that even though each of the Euclidean, max and absolute metric are essentially extensions of the absolute valued metric on the reals to "finite dimensional" Euclidean spaces, we must be careful when we try to further extend them to "infinite dimensional spaces". 

You can find notes for the above here.