  In this paper, we extend the G-expectation theory to infinite dimensions.
Such notions as a covariation set of G-normal distributed random variables,
viscosity solution, a stochastic integral driven by G-Brownian motion are
introduced and described in the given infinite dimensional case. We also give a
probabilistic representation of the unique viscosity solution to the fully
nonlinear parabolic PDE with unbounded first order term in Hilbert space in
terms of G-expectation theory.
