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We prove pathwise uniqueness for a class of stochastic differential equations(SDE) on a Hilbert space with cylindrical Wiener noise, whose nonlinear driftparts are sums of the sub-differential of a convex function and a bounded part.This generalizes a classical result by one of the authors to infinitedimensions. Our results also generalize and improve recent results by N.Champagnat and P. E. Jabin, proved in finite dimensions, in the case wheretheir diffusion matrix is constant and nondegenerate and their weaklydifferentiable drift is the (weak) gradient of a convex function. We also proveweak existence, hence obtain unique strong solutions by the Yamada-Watanabetheorem. The proofs are based in part on a recent maximal regularity result ininfinite dimensions, the theory of quasi-regular Dirichlet forms and aninfinite dimensional version of a Zvonkin-type transformation. As a mainapplication, we show pathwise uniqueness for stochastic reaction diffusionequations perturbed by a Borel measurable bounded drift. Hence, such SDE have aunique strong solution.

Strong uniqueness for SDEs in Hilbert spaces with nonregular drift