Summary: We consider the problem to minimize an integral functional defined on the space of absolutely continuous functions and measurable controls with values in an infinite-dimensional complex Banach space. 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. The paper is an extension of an earlier paper written for real Banach spaces. This extension covers optimal control problems which are governed by equations like the Schrödinger equation and other equations arising in quantum mechanics.