  In the high-energy quantum-physics literature one finds statements such as
"matrix algebras converge to the sphere". Earlier I provided a general precise
setting for understanding such statements, in which the matrix algebras are
viewed as quantum metric spaces, and convergence is with respect to a quantum
Gromov-Hausdorff-type distance.
  But physicists want even more to treat structures on spheres (and other
spaces), such as vector bundles, Yang-Mills functionals, Dirac operators, etc.,
and they want to approximate these by corresponding structures on matrix
algebras. In the present paper we treat this idea for vector bundles. We
develop a general precise way for understanding how, for two compact quantum
metric spaces that are close together, to a given vector bundle on one of them
there can correspond in a natural way a unique vector bundle on the other. We
then show explicitly how this works for the case of matrix algebras converging
to the 2-sphere.
