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Functional differentiability: Frechet, but not Hadamard?

  1. Apr 10, 2012 #1
    I have a question regarding functional differentiablility.

    I understand that Frechet differntiability of a functional T with respect to a norm [itex]\rho_1[/itex] 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 [itex]\rho[/itex], but that the same functional is Frechet differentiable with respect to a different norm [itex]\rho_2[/itex]. Especially, this turn out to be the case for some functionals when [itex]\rho_1(\Delta)=sup_x |\Delta(x)|[/itex] is the infinity norm, and
    [itex]\rho_2(\Delta)=\int |\Delta(x)|dx[/itex] is the L_1-norm.

    According to a number of sources this should be the case for some functionals on the quite simple form [itex]T(H)=s(\int x dH(x) )=s(\int x h(x)dx),[/itex]
    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.
     
  2. jcsd
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