The main objective of this work is to study the existence of Lagrange multipliers for infinite dimensional problems under Gateaux differentiability assumptions on the data. Our investigation follows two main steps: the proof of the existence of Lagrange multipliers under a calmness assumption on the constraints and the study of sufficient conditions, which only use the Gateaux derivative of the function defining the constraint, that ensure this assumption. We apply the abstract results to show directly the existence of Lagrange multipliers of two classes of standard stochastic optimal control problems.