Functional differentiability: Frechet, but not Hadamard?

[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.

 

[fresh_42]

Here are many examples, and proposition 4.1 is what you are looking for, i.e. contains the hint that R.M.Dudley (1992,1994) came up with that example. So that's where I would start to look for.
https://www.stat.washington.edu/people/jaw/COURSES/580s/581/LECTNOTES/ch7.pdf