Good news on metric spaces

I realized in a recent post on metric spaces that I had somewhat maligned them, dragged their good names through the mud.  In a spirit of reconciliation, I thought maybe I would mention some positive news about them.  These are some of the main tools that researchers use when investigating a metric space.

  1. Lipschitz functions. I mentioned that the notion of “addition” is not (necessarily) defined in a metric space, so neither is differentiability (though I defined a notion of a graph Laplacian… note that the graph Laplacian is a particular differential operator, defined on a particular metric space).  However, we do have a notion of Lipschitz functions, which stand in for differentiable functions.  A Lipschitz function is one where the distance between two points is not changed too much by the function.  Specifically, if f: X \to Y is a map between metric spaces X and Y, then f is Lipschitz if there is a constant C with  d_Y(f(x),f(y)) \leq C d_X(x,y) for all x,y in X.  Notice that if we had a good notion of a limit, we might take a limit as y gets close to x, and call C the derivative of f at x.  In fact, this is precisely what we do in a Euclidean space.  However, Lipschitz is a little weaker (i.e., easier to satisfy) than differentiability on compact sets.  For example, the absolute value function f(x) = |x| is Lipschitz with constant 1.  Notice, however, that a function like f(x) = x^2 is not Lipschitz on the real line, since the slope of f grows without bound.
  2. Extension functions. Given a Lipschitz map on a subset of a metric space, is there a Lipschitz map on the entire metric space whose restriction to the subset is the original map?  This is interesting to think of in light of boundary value problems in partial differential equations, as here we ask for any function with the given boundary data, and at least as much regularity as on the boundary.  The most famous theorem in this direction was proven by Kirszbraun in 1934, and says that if f is a map from a subset of a Hilbert space to another Hilbert space, then an extension exists with Lipschitz constant equal to the original.  In particular, any Lipschitz function on the real line extends to a Lipschitz function on the plane.  Note that one should not expect uniqueness, or else determining a Lipschitz map on a subset would determine it everywhere!
  3. Embedding theorems. For work that absolutely requires adding together elements, there exist embedding theorems.  One useful theorem states that any separable (i.e. has a countable dense subset) metric space may be isometrically embedded into a separable Banach space (namely, \ell^{\infty}(\mathbb{Z}), the space of bounded sequences) via the following process:  choose a countable dense subset \{ x_j \}_{j = 0}^{\infty}.  Then we define the isometry j by j(x) = (d(x,x_0) - d(x_0,x_0),d(x,x_1)-d(x_0,x_1),d(x,x_2)-d(x_0,x_2),\ldots)  This turns out to preserve the metric (i.e. be an isometry) while giving us a reasonable way of adding points and a way of reversing the operation.  A powerful tool is to prove results in the Banach space setting, then send the results back to the metric space world.  An interesting exercise is to explicitly construct this isometry for, say, the real line or the plane.  One quickly realizes how non-canonical the isometry is, as well as how hard it can be to write these maps down!
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