Heading off today for a conference on mapping theory in metric spaces. The hope is to generalize some results from my dissertation to metric spaces (rather than Euclidean spaces). I thought I might share some thoughts so that one can start thinking about the possible difficulties of analysis in metric spaces.
First, a metric space is just a set of points M together with a metric d that takes two points and returns a distance between them. The function d must have a few properties, which make intuitive sense: distance is a positive number, the distance from a to b must be the same as the distance from b to a, the distance from any point to itself is zero, and there must be a triangle inequality. The triangle inequality just says that there are no shortcuts: the distance from a to b is less than the distance from a to c plus the distance from c to b. ( )
An easy example is the plane, together with normal Euclidean distance. A great collection of an example that is not a vector space would be a graph (in the sense of a collection of nodes and lines connecting those nodes, rather than the graph of a function). In this case, the distance is the length of the shortest path between two nodes on the graph. This also displays some of the difficulties in working with metric spaces. In particular, given two points on a graph, there is no obvious way of adding those points.This problem becomes more pronounced when, for example, one tries define the derivative of a real valued function on a graph. A way of doing this is to try to keep important properties of the operator. For example, the Laplacian tends to smooth out a function, and so a reasonable definition will be to average a node with its connected nodes. More information can be found at this article. The gif on this page is a randomly generated graph being evolved according to the heat equation, with some random heat shocks thrown in. I’m not entirely happy with the figure, since the x-y coordinates are meaningless, but the z coordinates represent the heat of a node at time t.