Zero product preserving maps on \(C^1[0,1]\) (Q944341)

Let \(C^1[0,1]\) denote the space of continuously differentiable complex-valued functions equipped with the norm \(\| f\| = \| f\| _{\infty} + \| f'\| _{\infty}\). Let \(X\) be a Banach space. In this interesting paper, the authors give a complete description of bilinear maps \(\phi: C^1[0,1] \times C^1[0,1]\rightarrow X\) that preserve zero products, in the sense that \(fg=0 \Rightarrow \phi(f,g)=0\). They are of the form \(\phi(f,g)= P(fg)+Q(fg')+R(f'g')\) for some continuous linear maps \(P:C^1[0,1] \rightarrow X\) and \(Q,R:C[0,1]\rightarrow X\).