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 (which may be nonlinear) and the Gateaux derivative (which they take to be linear). 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 formula_1 and formula_2 are locally convex topological vector spaces (for example, Banach spaces), formula_3 is open, and formula_4. The Gateaux differential formula_5 of formula_6 at formula_7 in the direction formula_8 is defined as
If the limit exists for all formula_8, then one says that formula_6 is Gateaux differentiable at formula_11.
The limit appearing in () is taken relative to the topology of formula_2. If formula_1 and formula_2 are real topological vector spaces, then the limit is taken for real formula_15. On the other hand, if formula_1 and formula_2 are complex topological vector spaces, then the limit above is usually taken as formula_18 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 formula_7, the Gateaux differential defines a function
This function is homogeneous in the sense that for all scalars formula_21,
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 formula_23 if formula_1 and formula_2 are infinite dimensional. Furthermore, for Gateaux differentials that "are" linear and continuous in formula_23, there are several inequivalent ways to formulate their continuous differentiability.
For example, consider the real-valued function formula_6 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 formula_30. In infinite dimensions, any discontinuous linear functional on formula_1 is Gateaux differentiable, but its Gateaux differential at formula_32 is linear but not continuous.
If formula_6 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 formula_6 from a "complex" Banach space formula_1 to another complex Banach space formula_2, the Gateaux derivative (where the limit is taken over complex formula_15 tending to zero as in the definition of complex differentiability) is automatically linear, a theorem of . Furthermore, if formula_6 is (complex) Gateaux differentiable at each formula_7 with derivative
then formula_6 is Fréchet differentiable on formula_42 with Fréchet derivative formula_43 . 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 Gateaux differentiability may be defined in two inequivalent ways. Suppose that formula_44 is Gateaux differentiable at each point of the open set formula_42. One notion of continuous differentiability in formula_42 requires that the mapping on the product space
be continuous. Linearity need not be assumed: if formula_1 and formula_2 are Fréchet spaces, then formula_50 is automatically bounded and linear for all formula_11 .
A stronger notion of continuous differentiability requires that
be a continuous mapping
from formula_42 to the space of continuous linear functions from formula_1 to formula_2. Note that this already presupposes the linearity of formula_57.
As a matter of technical convenience, this latter notion of continuous differentiability is typical (but not universal) when the spaces formula_1 and formula_2 are Banach, since formula_60 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 formula_61, higher order Gateaux derivative cannot be defined in this way. Instead the formula_62th order Gateaux derivative of a function formula_63 in the direction formula_64 is defined by
Rather than a multilinear function, this is instead a homogeneous function of degree formula_62 in formula_64.
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 formula_6, at least in the special case where formula_6 is scalar-valued. However, this may fail to have any reasonable properties at all, aside from being separately homogeneous in formula_64 and formula_70. It is desirable to have sufficient conditions in place to ensure that formula_71 is a symmetric bilinear function of formula_64 and formula_70, and that it agrees with the polarization of formula_74.
For instance, the following sufficient condition holds . Suppose that formula_6 is formula_76 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 formula_71 is bilinear and symmetric in formula_64 and formula_70. By virtue of the bilinearity, the polarization identity holds
relating the second order derivative formula_83 with the differential formula_84. Similar conclusions hold for higher order derivatives.
Properties.
A version of the fundamental theorem of calculus holds for the Gateaux derivative of formula_6, provided formula_6 is assumed to be sufficiently continuously differentiable. Specifically:
Many of the other familiar properties of the derivative follow from this, such as multilinearity and commutativity of the higher-order derivatives. Further properties, also consequences of the fundamental theorem, include:
Example.
Let formula_1 be the Hilbert space of square-integrable functions on a Lebesgue measurable set formula_104 in the Euclidean space formula_105. The functional
where formula_6 is a real-valued function of a real variable and formula_11 is defined on formula_104 with real values, has Gateaux derivative
Indeed, the above is the limit formula_18 of