Hello,

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:

$\displaystyle (f*g)(x+h)=(f(x+h))*(g(x+h))=( f(x)+df(x)[h]+o(\|h\|) )*( g(x)+dg(x)[h]+o(\|h\|) )

=...=f(x)*g(x)+f(x)*dg(x)[h]+df(x)[h]*g(x)+o(\|h\|)$

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 $\displaystyle \|u*v\|<=\|u\|*\|v\|$

Thank you a lot!

Regards