  We investigate the linear quadratic stochastic optimal control problems in
infinite dimension without Markovian restriction for coefficients. The
necessary and sufficient conditions for open-loop optimal controls are
presented. We prove the Fr\'echet differentiable of the cost functional with
respect to the control variable, and the Fr\'echet derivatives are
characterized in detail by operators derived from dual analysis, which are
proven to be the stationary conditions. Transposition methods are adopted to
deal with the adjoint equations. As applications, we employ the results to
study open-loop Nash equilibria for two-person stochastic differential games.
