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.
