We consider the problem to minimize an integral functional defined on the space of absolutely continuous functions and measurable control functions with values in infinite dimensional real Banach spaces. The states are governed by abstract first order semilinear differential equations and are subject to periodic or anti-periodic type boundary conditions. We derive necessary conditions for optimality and introduce the notion of a dual field of extremals to obtain sufficient conditions for optimality. Such a dual field of extremals is constructed and a dual optimal synthesis is proposed.