  In this paper, we study the maximum principle of mean field type control
problems when the volatility function depends on the state and its measure and
also the control, by using our recently developed method. Our method is to
embed the mean field type control problem into a Hilbert space to bypass the
evolution in the Wasserstein space. We here give a necessary condition and a
sufficient condition for these control problems in Hilbert spaces, and we also
derive a system of forward-backward stochastic differential equations.
