Functional differentiability: Frechet, but not Hadamard?

In summary, Frechet differentiability measures the overall sensitivity of a function's output to its input, while Hadamard differentiability measures the local sensitivity to small changes in input. Understanding this distinction is important for understanding function behavior and choosing the appropriate type of differentiability for a problem. A function can be Frechet differentiable but not Hadamard differentiable, such as the absolute value function and x^3 at x=0. Frechet differentiability is closely related to continuity, as a function that is Frechet differentiable at a point is also continuous at that point.
  • #1
Testguy
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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.
 
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  • #3


Hello,

Thank you for your question regarding functional differentiability. You are correct in your understanding that Frechet differentiability of a functional T with respect to a norm \rho_1 implies Hadamard differentiability with respect to the same norm. However, as you mentioned, there can be cases where a functional is not Hadamard differentiable with respect to a certain norm, but is Frechet differentiable with respect to a different norm. In particular, this can occur when using the infinity norm and the L_1-norm.

As for your question about finding a function s where this is the case, I believe I have an example for you. Consider the functional T(H) = \int_a^b x^2 dH(x) with H being a probability distribution on the interval [a,b]. In this case, the function h(x) = 2x and s(x) = x^2 satisfy the conditions you mentioned.

I hope this helps and please let me know if you have any further questions. Best of luck with your studies!
 

Related to Functional differentiability: Frechet, but not Hadamard?

1. What is the difference between Frechet and Hadamard differentiability?

Frechet differentiability is a type of differentiability that measures the sensitivity of a function's output to its input, while Hadamard differentiability measures the sensitivity of a function's output to small changes in its input. In other words, Frechet differentiability focuses on the overall behavior of a function, while Hadamard differentiability looks at the local behavior.

2. Why is it important to distinguish between Frechet and Hadamard differentiability?

Understanding the difference between Frechet and Hadamard differentiability is important because it can help us understand the behavior and properties of different types of functions. It can also help us identify which type of differentiability is more appropriate for a particular problem or application.

3. Can a function be Frechet differentiable but not Hadamard differentiable?

Yes, a function can be Frechet differentiable but not Hadamard differentiable. This means that the function is sensitive to changes in its input, but not necessarily to small changes in its input. In other words, the function may not have a well-defined local behavior.

4. What are some examples of functions that are Frechet differentiable but not Hadamard differentiable?

Some examples of functions that are Frechet differentiable but not Hadamard differentiable include the absolute value function and the function x^3 at x=0. These functions exhibit sensitivity to changes in their input, but not to small changes in their input.

5. How is Frechet differentiability related to the concept of continuity?

Frechet differentiability is closely related to the concept of continuity. A function that is Frechet differentiable at a point is also continuous at that point. This means that as the input approaches a particular value, the output of the function also approaches a particular value. However, the reverse is not necessarily true – a function can be continuous at a point but not Frechet differentiable at that point.

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