In a framework of a differential game with a large number N of indistinguishable players, the intricate structure of the Nash system can be approximated, for N large with an easier system of coupled PDEs, called Mean Field Games system. It consists of a backward HJB equation for the single agent’s value function, coupled with a forward FP equation for the distribution of the population. However, a rigorous proof of this convergence passes through the study of a PDE equation in the space of measures, called Master Equation. The novelty we introduce is to study such PDE system in the case where the state variable lives in an Hilbert space. In control theory, such problems naturally arise when the agent’s dynamics depend on additional variables beyond time, such as age, spatial position, or path-dependent effects. We study the case when the cost function is quadratic and the dependence on the distribution enters in the objective functional and in the dynamics through the mean. Here, the particular structure of the coefficients and the data of the problem allows us to give a precise representation of the solutions for the Nash system, the Mean Field Games and the Master Equation. This representation allows to pass to the limit for $N\to +\infty$ and to prove the desired convergence result. Finally we apply our results to a vintage capital model and to a systemic risk model with delay.