Let and be Banach spaces and let be a function between them. is said to be Gâteaux differentiable if there exists an operator such that, for all ,
(1)
|
The operator is called the Gâteaux derivative of at . 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 is Gâteaux differentiable at a point if and only if all the directional operators
(2)
|
exist and form a bounded linear operator . In addition, the Gâteaux derivative satisfies analogues of many properties from basic calculus including a mean-value property of the form
(3)
|
One definition of the Fréchet derivative pertains to uniform existence of the Gâteaux derivative on the unit sphere of (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.