  In a previous paper, \cite{Berndtsson}, we have studied a property of
subharmonic dependence on a parameter of Bergman kernels for a family of
weighted $L^2$-spaces of holomorphic functions. Here we prove a result on the
curvature of a vector bundle defined by this family of $L^2$-spaces itself,
which has the earlier results on Bergman kernels as a corollary. Applying the
same arguments to spaces of holomorphic sections to line bundles over a locally
trivial fibration we also prove that if a holomorphic vector bundle, $V$, over
a complex manifold is ample in the sense of Hartshorne, then $V\gr\det V$ has
an Hermitian metric with curvature strictly positive in the sense of Nakano.
