Gâteaux derivative

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Gâteaux 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 Tikhomirov (2001), 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

Supposeandarelocally convextopological vector spaces(for example,Banach spaces),is open, and. The Gateaux differentialofatin the directionis defined as

**(1)**

If the limit exists for all, then one says thatis Gateaux differentiable at.

The limit appearing in (**1**) is taken relative to the topology of. Ifandarerealtopological vector spaces, then the limit is taken for real. On the other hand, ifandarecomplextopological vector spaces, then the limit above is usually taken asin thecomplex planeas in the definition ofcomplex differentiability. In some cases, aweak limitis 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 function

This 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 theFréchet derivative. Even if linear, it may fail to depend continuously onifandare infinite dimensional. Furthermore, for Gateaux differentials that are linear and continuous in, there are several inequivalent ways to formulate theircontinuous differentiability.

For example, consider the real-valued functionof two real variables defined by

This is Gateaux differentiable at (0, 0), with its differential there being

However this is continuous but not linear in the arguments. In infinite dimensions, anydiscontinuous linear functionalonis Gateaux differentiable, but its Gateaux differential atis linear but not continuous.

Relation with the Fréchet derivative

Ifis 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 functionsfrom aBanach spaceto another complex Banach space, the Gateaux derivative (where the limit is taken over complextending to zero as in the definition ofcomplex differentiability) is automatically linear, a theorem ofZorn (1945). Furthermore, ifis (complex) Gateaux differentiable at eachwith derivative

thenis Fréchet differentiable onwith Fréchet derivative(Zorn 1946). This is analogous to the result from basiccomplex analysisthat a function isanalyticif it is complex differentiable in an open set, and is a fundamental result in the study ofinfinite dimensional holomorphy.

Continuous differentiability

Continuous Gateaux differentiability may be defined in two inequivalent ways. Suppose thatis Gateaux differentiable at each point of the open set. One notion of continuous differentiability inrequires that the mapping on theproduct space

becontinuous. Linearity need not be assumed: ifandare Fréchet spaces, thenis automatically bounded and linear for all(Hamilton 1982).

A stronger notion of continuous differentiability requires that

be a continuous mapping

fromto the space of continuous linear functions fromto. Note that this already presupposes the linearity of.

As a matter of technical convenience, this latter notion of continuous differentiability is typical (but not universal) when the spacesandare Banach, sinceis 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 spaceshas applications such as theNash–Moser inverse function theoremin which the function spaces of interest often consist ofsmooth functionson amanifold.

Higher derivatives

Whereas higher order Fréchet derivatives are naturally defined asmultilinear functionsby iteration, using the isomorphisms, higher order Gateaux derivative cannot be defined in this way. Instead theth order Gateaux derivative of a functionin the directionis defined by

**(2)**

Rather than a multilinear function, this is instead ahomogeneous functionof degreein.

There is another candidate for the definition of the higher order derivative, the function

**(3)**

that arises naturally in the calculus of variations as the second variation of, at least in the special case whereis scalar-valued. However, this may fail to have any reasonable properties at all, aside from being separately homogeneous inand. It is desirable to have sufficient conditions in place to ensure thatis a symmetric bilinear function ofand, and that it agrees with thepolarizationof.

For instance, the following sufficient condition holds (Hamilton 1982). Suppose thatisin the sense that the mapping

is continuous in the product topology, and moreover that the second derivative defined by (3) is also continuous in the sense that

is continuous. Thenis bilinear and symmetric inand. By virtue of the bilinearity, the polarization identity holds

relating the second order derivativewith the differential. Similar conclusions hold for higher order derivatives.

Properties

A version of thefundamental theorem of calculusholds for the Gateaux derivative of, providedis 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 ,

where the integral is theGelfand–Pettis integral(the weak integral).

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:

(The chain rule)

for alland. (Note well that, as with simplepartial derivatives, the Gateaux derivative does not satisfy the chain rule if the derivative is permitted to be discontinuous.)