This paper deals with necessary conditions for optimization problems with infinitely many inequality constraints assuming various differentiability conditions. By introducing a second topologyNon a topological vector space we define generalized versions of differentiability and tangential cones. Different choices ofNlead to Gâteaux-, Hadamaed- and weak differentiability with corresponding tangential cones. The general concept is used to derive necessary conditions for local optimal points in form of inequalities and generalized multiplier rules, Special versions of these theorems are obtained for different differentiability assumptions by choosing properly. An application to approximation theory is given.