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 at age 25 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  and  are locally convex topological vector spaces (for example,  Banach spaces),  is open, and   The Gateaux differential  of  at  in the direction  is defined as

If 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 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 the Fréchet derivative.  Even if linear, it may fail to depend continuously on  if  and  are infinite dimensional.  Furthermore, for Gateaux differentials that  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  Banach space  to another complex Banach space  the Gateaux derivative (where the limit is taken over complex  tending to zero as in the definition of complex differentiability) is automatically linear, a theorem of .  Furthermore, if  is (complex) 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 (but not universal) 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 by

Rather 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  where the integral is the Gelfand–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 all  and   (Importantly, as with simple partial derivatives, the Gateaux derivative does  satisfy the chain rule if the derivative is permitted to be discontinuous.)

 (Taylor's theorem with remainder)Suppose that the line segment between  and  lies entirely within   If  is  then  where the remainder term is given by

Example

Let  be the Hilbert space of square-integrable functions on a Lebesgue measurable set  in the Euclidean space  The functional

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

See also

References

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Generalizations of the derivative
Topological vector spaces