  We construct an analytic multiplicative model of smooth K-theory. We further
introduce the notion of a smooth K-orientation of a proper submersion and
define the associated push-forward which satisfies functoriality, compatibility
with pull-back diagrams, and projection and bordism formulas. We construct a
multiplicative lift of the Chern character from smooth K-theory to smooth
rational cohomology and verify that the cohomological version of the
Atiyah-Singer index theorem for families lifts to smooth cohomology.
