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 fg=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.