  We prove a large deviation principle (LDP) for a general class of Banach
space valued stochastic differential equations (SDE) that is uniform with
respect to initial conditions in bounded subsets of the Banach space. A key
step in the proof is showing that a uniform large deviation principle over
compact sets is implied by a uniform over compact sets Laplace principle.
Because bounded subsets of infinite dimensional Banach spaces are in general
not relatively compact in the norm topology, we embed the Banach space into its
double dual and utilize the weak-$\star $ compactness of closed bounded sets in
the double dual space. We prove that a modified version of our stochastic
differential equation satisfies a uniform Laplace principle over weak-$\star $
compact sets and consequently a uniform over bounded sets large deviation
principle. We then transfer this result back to the original equation using a
contraction principle. The main motivation for this uniform LDP is to
generalize results of Freidlin and Wentzell concerning the behavior of finite
dimensional SDEs. Here we apply the uniform LDP to study the asymptotics of
exit times from bounded sets of Banach space valued small noise SDE, including
reaction diffusion equations with multiplicative noise and $2$-dimensional
stochastic Navier-Stokes equations with multiplicative noise.
