  We prove pathwise (hence strong) uniqueness of solutions to stochastic
evolution equations in Hilbert spaces with merely measurable bounded drift and
cylindrical Wiener noise, thus generalizing Veretennikov's fundamental result
on $\mathbb{R}^d$ to infinite dimensions. Because Sobolev regularity results
implying continuity or smoothness of functions do not hold on
infinite-dimensional spaces, we employ methods and results developed in the
study of Malliavin-Sobolev spaces in infinite dimensions. The price we pay is
that we can prove uniqueness for a large class, but not for every initial
distribution. Such restriction, however, is common in infinite dimensions.
