The Fréchet derivative of a map 
at 
, where 
is an open subset
of a normed vector space 
and 
is a normed vector
space, is the unique bounded linear transformation 
, when it exists, such that
 |
In that case, 
is Fréchet differentiable at 
and 
is denoted 
. For functions of one real variable or complex
variable, this definition reduces to the usual derivative.
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 
is Fréchet differentiable iff the limit used to describe the
Gâteaux derivative exists uniformly
with respect to vectors 
on the unit sphere of the domain space 
; as such, this uniform limit (when
it exists) is called the Fréchet derivative (Andrews and Hopper 2011).
