Named after Maurice Fréchet, 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 of variations.
Generally, it extends the idea of the derivative from real-valued functions of one real variable to functions on normed spaces.
The Fréchet derivative has applications to nonlinear problems throughout mathematical analysis and physical sciences, particularly to the calculus of variations and much of nonlinear analysis and nonlinear functional analysis.
The limit here is meant in the usual sense of a limit of a function defined on a metric space (see Functions on metric spaces), using
as the two metric spaces, and the above expression as the function of argument
Equivalently, the first-order expansion holds, in Landau notation
denotes the space of all bounded linear operators from
(which is assumed; bounded and continuous are equivalent).
is a scalar (a real or complex number), then the Fréchet derivative obeys the following properties:
is Fréchet differentiable and yet fails to have continuous partial derivatives at
One of the simplest (nontrivial) examples in infinite dimensions, is the one where the domain is a Hilbert space (
Using continuity of the norm and inner product we obtain:
the norm is not differentiable, that is, there does not exist bounded linear functional
Riesz Representation Theorem tells us that
This is analogous to the fact that the existence of all directional derivatives at a point does not guarantee total differentiability (or even continuity) at that point.
is not a linear operator, so this function is not Fréchet differentiable.
These cases can occur because the definition of the Gateaux derivative only requires that the difference quotients converge along each direction individually, without making requirements about the rates of convergence for different directions.
Thus, for a given ε, although for each direction the difference quotient is within ε of its limit in some neighborhood of the given point, these neighborhoods may be different for different directions, and there may be a sequence of directions for which these neighborhoods become arbitrarily small.
If a sequence of points is chosen along these directions, the quotient in the definition of the Fréchet derivative, which considers all directions at once, may not converge.
Thus, in order for a linear Gateaux derivative to imply the existence of the Fréchet derivative, the difference quotients have to converge uniformly for all directions.
is a differentiable function at all points in an open subset
To make it easier to work with second-order derivatives, the space on the right-hand side is identified with the Banach space
again, to obtain the third order derivative, which at each point will be a trilinear map, and so on.
taking values in the Banach space of continuous multilinear maps in
In this section, we extend the usual notion of partial derivatives which is defined for functions of the form
be Banach spaces (over the same field of scalars), and let
The notion of the Fréchet derivative can be generalized to arbitrary topological vector spaces (TVS)
if there exists a continuous linear operator
(Lang p. 6) If the Fréchet derivative exists then it is unique.
A function that is Fréchet differentiable at a point is necessarily continuous there and sums and scalar multiples of Fréchet differentiable functions are differentiable so that the space of functions that are Fréchet differentiable at a point form a subspace of the functions that are continuous at that point.