Problems of optimal control in Banach spaces are transformed in optimization problems, in which the directional derivatives of the operators and functionals satisfy some conditions of convexity and optimality. Necessary optimality conditions are derived by separation of a closed convex cone from a point outside of itself. The condition of “complementary slackness” is not necessary. The mentioned cone need not to possess inner points. The results are applied to the given control problem and are formulated as maximum principle. Examples are given.