In mathematics, the Fréchet derivative is a derivative defined on normed spaces. 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 should be contrasted to the more general Gateaux derivative which is a generalization of the classical directional derivative.
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.
Definition
Let and be normed vector 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 (see Functions on metric spaces), using and as the two metric spaces, and the above expression as the function of argument in As a consequence, it must exist for all sequences of non-zero elements of that converge to the zero vector Equivalently, the first-order expansion holds, in Landau notation
If there exists such an operator it is unique, so we write and call it the Fréchet derivative of at A function that is Fréchet differentiable for any point of is said to be C1 if the function
is continuous ( denotes the space of all bounded linear operators from to ). Note that this is not the same as requiring that the map be continuous for each value of (which is assumed; bounded and continuous are equivalent).
This notion of derivative is a generalization of the ordinary derivative of a function on the real numbers since the linear maps from to are just multiplication by a real number. In this case, is the function
Properties
A function differentiable at a point is continuous at that point.
Differentiation is a linear operation in the following sense: if and are two maps which are differentiable at and is a scalar (a real or complex number), then the Fréchet derivative obeys the following properties:
The chain rule is also valid in this context: if is differentiable at and is differentiable at then the composition is differentiable in and the derivative is the composition of the derivatives:
Finite dimensions
The Fréchet derivative in finite-dimensional spaces is the usual derivative. In particular, it is represented in coordinates by the Jacobian matrix.
Suppose that is a map, with an open set. If is Fréchet differentiable at a point then its derivative is
where denotes the Jacobian matrix of at
Furthermore, the partial derivatives of are given by
where is the canonical basis of Since the derivative is a linear function, we have for all vectors that the directional derivative of along is given by
If all partial derivatives of exist and are continuous, then is Fréchet differentiable (and, in fact, C1). The converse is not true; the function
is Fréchet differentiable and yet fails to have continuous partial derivatives at
Example in infinite dimensions
One of the simplest (nontrivial) examples in infinite dimensions, is the one where the domain is a Hilbert space () and the function in interest is the norm. So consider
First assume that Then we claim that the Fréchet derivative of at is the linear functional defined by
Indeed,
Using continuity of the norm and inner product we obtain:
As and because of the Cauchy-Schwarz inequality
is bounded by thus the whole limit vanishes.
Now we show that at the norm is not differentiable, that is, there does not exist bounded linear functional such that the limit in question to be Let be any linear functional. Riesz Representation Theorem tells us that could be defined by for some Consider
In order for the norm to be differentiable at we must have
We will show that this is not true for any If obviously independently of hence this is not the derivative. Assume If we take tending to zero in the direction of (that is, where ) then hence
(If we take tending to zero in the direction of we would even see this limit does not exist since in this case we will obtain ).
The result just obtained agrees with the results in finite dimensions.