Gateaux derivative
In mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René Gateaux, a French mathematician who died young in World War I, it is defined for functions between locally convex topological vector spaces such as Banach spaces. Like the Fréchet derivative on a Banach space, the Gateaux differential is often used to formalize the functional derivative commonly used in the calculus of variations and physics.
Unlike other forms of derivatives, the Gateaux differential of a function may be nonlinear. However, often the definition of the Gateaux differential also requires that it be a continuous linear transformation. Some authors, such as, draw a further distinction between the Gateaux differential and the Gateaux derivative. In most applications, continuous linearity follows from some more primitive condition which is natural to the particular setting, such as imposing complex differentiability in the context of infinite dimensional holomorphy or continuous differentiability in nonlinear analysis.
Definition
Suppose and are locally convex topological vector spaces, is open, and. The Gateaux differential of at in the direction is defined asIf the limit exists for all, then one says that is Gateaux differentiable at.
The limit appearing in is taken relative to the topology of. If and are real topological vector spaces, then the limit is taken for real. On the other hand, if and are complex topological vector spaces, then the limit above is usually taken as in the complex plane as in the definition of complex differentiability. In some cases, a weak limit is taken instead of a strong limit, which leads to the notion of a weak Gateaux derivative.
Linearity and continuity
At each point, the Gateaux differential defines a functionThis function is homogeneous in the sense that for all scalars,
However, this function need not be additive, so that the Gateaux differential may fail to be linear, unlike the Fréchet derivative. Even if linear, it may fail to depend continuously on if and are infinite dimensional. Furthermore, for Gateaux differentials that are linear and continuous in, there are several inequivalent ways to formulate their continuous differentiability.
For example, consider the real-valued function of two real variables defined by
This is Gateaux differentiable at, with its differential there being
However this is continuous but not linear in the arguments. In infinite dimensions, any discontinuous linear functional on is Gateaux differentiable, but its Gateaux differential at is linear but not continuous.
;Relation with the Fréchet derivative
If is Fréchet differentiable, then it is also Gateaux differentiable, and its Fréchet and Gateaux derivatives agree. The converse is clearly not true, since the Gateaux derivative may fail to be linear or continuous. In fact, it is even possible for the Gateaux derivative to be linear and continuous but for the Fréchet derivative to fail to exist.
Nevertheless, for functions from a complex Banach space to another complex Banach space, the Gateaux derivative is automatically linear, a theorem of. Furthermore, if is Gateaux differentiable at each with derivative
then is Fréchet differentiable on with Fréchet derivative . This is analogous to the result from basic complex analysis that a function is analytic if it is complex differentiable in an open set, and is a fundamental result in the study of infinite dimensional holomorphy.
;Continuous differentiability
Continuous Gateaux differentiability may be defined in two inequivalent ways. Suppose that is Gateaux differentiable at each point of the open set. One notion of continuous differentiability in requires that the mapping on the product space
be continuous. Linearity need not be assumed: if and are Fréchet spaces, then is automatically bounded and linear for all .
A stronger notion of continuous differentiability requires that
be a continuous mapping
from to the space of continuous linear functions from to. Note that this already presupposes the linearity of.
As a matter of technical convenience, this latter notion of continuous differentiability is typical when the spaces and are Banach, since is also Banach and standard results from functional analysis can then be employed. The former is the more common definition in areas of nonlinear analysis where the function spaces involved are not necessarily Banach spaces. For instance, differentiation in Fréchet spaces has applications such as the Nash–Moser inverse function theorem in which the function spaces of interest often consist of smooth functions on a manifold.
Higher derivatives
Whereas higher order Fréchet derivatives are naturally defined as multilinear functions by iteration, using the isomorphisms, higher order Gateaux derivative cannot be defined in this way. Instead the th order Gateaux derivative of a function in the direction is defined byRather than a multilinear function, this is instead a homogeneous function of degree in.
There is another candidate for the definition of the higher order derivative, the function
that arises naturally in the calculus of variations as the second variation of, at least in the special case where is scalar-valued. However, this may fail to have any reasonable properties at all, aside from being separately homogeneous in and. It is desirable to have sufficient conditions in place to ensure that is a symmetric bilinear function of and, and that it agrees with the polarization of.
For instance, the following sufficient condition holds. Suppose that is in the sense that the mapping
is continuous in the product topology, and moreover that the second derivative defined by is also continuous in the sense that
is continuous. Then is bilinear and symmetric in and. By virtue of the bilinearity, the polarization identity holds
relating the second order derivative with the differential. Similar conclusions hold for higher order derivatives.
Properties
A version of the fundamental theorem of calculus holds for the Gateaux derivative of, provided is assumed to be sufficiently continuously differentiable. Specifically:- Suppose that is in the sense that the Gateaux derivative is a continuous function. Then for any and,
Example
where is a real-valued function of a real variable and is defined on with real values, has Gateaux derivative
Indeed, the above is the limit of