Fréchet Derivative

A function f is Fréchet differentiable at a if

exists. This is equivalent to the statement that phi has a removable discontinuity at a, where

In literature, the Fréchet derivative is sometimes known as the strong derivative (Ostaszewski 2012) and can be seen as a generalization of the gradient to arbitrary vector spaces (Long 2009).

Every function which is Fréchet differentiable is both Carathéodory differentiable and Gâteaux differentiable. The relationship between the Fréchet derivative and the Gâteaux derivative can be made even more explicit by noting that a function f is Fréchet differentiable if and only if the limit used to describe the Gâteaux derivative exists uniformly with respect to vectors v on the unit sphere of the domain space X; as such, this uniform limit (when it exists) is what's called the Fréchet Derivative (Andrews and Hopper 2011)

See also

Portions of this entry contributed by Christopher Stover

Explore with Wolfram|Alpha

References

Referenced on Wolfram|Alpha

Cite this as:

Stover, Christopher and Weisstein, Eric W. "Fréchet Derivative." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FrechetDerivative.html

Subject classifications