# Functional differentiability: Frechet, but not Hadamard?

1. Apr 10, 2012

### Testguy

I have a question regarding functional differentiablility.

I understand that Frechet differntiability of a functional T with respect to a norm $\rho_1$ implies Hadamard differentiability of the functional T with respect to the same norm.

However, it is no surprise that there would be cases where a functional T is not Hadamard differentiable with respect to a norm $\rho$, but that the same functional is Frechet differentiable with respect to a different norm $\rho_2$. Especially, this turn out to be the case for some functionals when $\rho_1(\Delta)=sup_x |\Delta(x)|$ is the infinity norm, and
$\rho_2(\Delta)=\int |\Delta(x)|dx$ is the L_1-norm.

According to a number of sources this should be the case for some functionals on the quite simple form $T(H)=s(\int x dH(x) )=s(\int x h(x)dx),$
where h(x) is the usual derivative of the function H(x), and s is some differentiable function.

I do however not find any s where this is the case.

Can anyone help me out with such a function s?

Any help is appreciated.