i asked myself, whether the product rule is correct for functions which have values in a banach algebra.
I think it is true, but i'm not so sure. Do you know it? Or can you check my "proof" for the product rule please?:
Let f,g:V->W be two differentiable maps, whereas W is some banachalgebra. Then we have:
therefore we have the derivative:
f(x)*dg(x)[h]+df(x)[h]*g(x) and the product rule is proved.
Of course we use in the calculation above, that the multiplication is distributive and also that
Thank you a lot!