Interesting representations of a Lie group G often arise on spaces of functions on a manifold M where G acts. Such function spaces are usually infinite-dimensional. A fundamental problem is to find such geometric realizations of an abstract representation of G. A natural first step is to figure out what the dimension of the manifold ought to be. Another way to say this is that we would like to determine the dimension of a manifold by looking at a vector space of functions on it. There is an immediate problem. Because all separable Hilbert spaces are isomorphic, the space of square-integrable functions can tell us only whether or not M is infinite. One might hope that a more subtle space like smooth functions could do better, but again we are disappointed: the spaces of smooth functions on infinite compact manifolds are all isomorphic as topological vector spaces. One simple resolution of this problem comes from analysis. If M is a compact Riemannian manifold, then the Hilbert space L^2(M) carries a Laplace operator D. The asymptotic distibution of the eigenvalues of D is controlled by the dimension of M. Explicitly, the number of eigenvalues of D smaller than a large positive number T grows like T^{dim M/2}. I will explain how to implement this idea in representation theory, and a number of other ways to get at the same invariant.