The Fréchet derivative is a derivative defined on Banach spaces, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued function of multiple real variables, and to define the functional derivative used widely in the calculus variations.
Definition Let and be Banach spaces, and be an open subset of . A function is called Fréchet differentiable at if there exists a bounded linear operator such that
The limit here is meant in the usual sense of a limit of a function defined on a metric space.
Relation to the Gâteaux derivative
Definition A function is called Gâteaux differentiable at if has a directional derivative along all direction at . This means that the limit
exists for any choosen vector in , where is is from the scalar filed associated with .
If is Fréchet differentiable at , is is also Gâteaux differentiable there, and the limit is just .
If is a differentiable function at all points in an open subset of , it follows that its derivative
is a function from to the space of all bounded liner operators from to . This function may also have a derivative, the second order derivative of , which, by the definition of derivative, will be a map
To make it easier to work with second order derivatives, the space on the right-hand side is identified with the Banach space of all continuous bilinear map from to . An element in is thus identified with in such that for all and in