The Gâteaux derivative 
of 
at 
, for a function 
between Banach spaces 
and 
, is a bounded linear
operator 
such that, for all 
,
 |
(1)
|
The function 
is Gâteaux differentiable when such an operator
exists. Some authors allow an unbounded linear operator,
or use the term for the existence of all directional
derivatives without assuming linearity. The bounded linear convention is used
here.
If the Gâteaux derivative exists, it is unique.
A basic result about Gâteaux derivatives is that 
is Gâteaux differentiable at a point 
iff all the directional
derivatives
 |
(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)
|
provided 
is Gâteaux differentiable on the segment joining 
and 
, where the last norm is the operator
norm.
One definition of the Fréchet derivative pertains to uniform existence of the Gâteaux derivative
on the unit sphere of 
(Andrews and Hopper 2011). 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 differently in the case of infinite-dimensional space than in the
finite-dimensional case.
