  In this paper, the distribution dependent stochastic differential equation in
a separable Hilbert space with a Dini continuous drift is investigated. The
existence and uniqueness of weak and strong solutions are obtained. Moreover,
some regularity results as well as gradient estimates and log-Harnack
inequality are derived for the associated semigroup. In addition, dimensional
free Harnack inequality with power and shift Harnack inequality are also proved
when the noise is additive. All of the results extend the ones in the
distribution independent situation.
