Zero product preserving maps on C1[0, 1]✩

The main result of the paper characterizes continuous bilinear maps φ from C1[0, 1] × C1[0, 1] into a Banach space X with the property that φ( f , g) = 0 whenever f g = 0. This is applied to the study of zero product preserving operators on C1[0, 1], and operators on C1[0, 1] satisfying a version of the condition of the locality of an operator.