  We consider $C^2$ Fr\'echet differentiable mappings of Banach spaces leaving
invariant compactly supported Borel probability measures, and study the
relation between entropy and volume growth for a natural notion of volume
defined on finite dimensional subspaces. SRB measures are characterized as
exactly those measures for which entropy is equal to volume growth on unstable
manifolds, equivalently the sum of positive Lyapunov exponents of the map. In
addition to numerous difficulties incurred by our infinite-dimensional setting,
a crucial aspect to the proof is the technical point that the volume elements
induced on unstable manifolds are regular enough to permit distortion control
of iterated determinant functions. The results here generalize previously known
results for diffeomorphisms of finite dimensional Riemannian manifolds, and are
applicable to dynamical systems defined by large classes of dissipative
parabolic PDEs.
