TOPICS
Search

Gâteaux Derivative


Let X and Y be Banach spaces and let f:X->Y be a function between them. f is said to be Gâteaux differentiable if there exists an operator T_x:X->Y such that, for all v in X,

 lim_(t->0)(f(x+tv)-f(x))/t=T_xv.
(1)

The operator T_x is called the Gâteaux derivative of f at x. T_x is sometimes assumed to be bounded, though much of the theory of Gâteaux differentiability remains unchanged without this assumption.

If the Gâteaux derivative exists, it is unique.

A basic result about Gâteaux derivatives is that f is Gâteaux differentiable at a point x in X if and only if all the directional operators

 delta_vf(x)=d/(dt)|_(t=0)f(x+tv)
(2)

exist and form a bounded linear operator Df(x):v|->delta_vf(x). In addition, the Gâteaux derivative satisfies analogues of many properties from basic calculus including a mean-value property of the form

 |f(y)-f(x)|<=|x-y|sup_(0<=theta<=1)|Df(thetax+(1-theta)y)|.
(3)

One definition of the Fréchet derivative pertains to uniform existence of the Gâteaux derivative on the unit sphere of X (Andrews and Hopper). In particular, then, Fréchet differentiability is stronger than differentiability in the Gâteaux sense, meaning that every function which is Fréchet differentiable is automatically differentiable in the sense of Gâteaux, though the converse fails in general. Andrews and Hopper give some criteria for when the notions are equivalent while noting that the two notions behave drastically different in the case of infinite-dimensional space than in the finite-dimensional case.


See also

Banach Space, Bounded Operator, Derivative, Differentiable, Directional Derivative, Fréchet Derivative, Mean-Value Theorem

This entry contributed by Christopher Stover

Explore with Wolfram|Alpha

References

Andrews, B. and Hopper, C. The Ricci Flow in Riemannian Geometry. Berlin: Springer, 2011.

Cite this as:

Stover, Christopher. "Gâteaux Derivative." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/GateauxDerivative.html

Subject classifications