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 map 
, when it exists, such that
 |
In that case, 
is Fréchet differentiable at 
and 
is denoted 
. For functions of one real 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).
