Gâteaux derivative
Encyclopedia
In mathematics
, the Gâteaux differential or Gâteaux derivative is a generalization of the concept of directional derivative
in differential calculus
. Named after René Gâteaux
, a French mathematician who died young in World War I
, it is defined for functions between locally convex topological vector space
s such as Banach space
s. Like the Fréchet derivative
on a Banach space, the Gâteaux differential is often used to formalize the functional derivative
commonly used in the calculus of variations
and physics
.
Unlike other forms of derivatives, the Gâteaux differential of a function may be nonlinear. However, often the definition of the Gâteaux differential also requires that it be a continuous linear transformation. Some authors, such as , draw a further distinction between the Gâteaux differential (which may be nonlinear) and the Gâteaux 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.
s (for example, Banach space
s), U ⊂ X is open, and F : X → Y. The Gâteaux differential dF(u;ψ) of F at u ∈ U in the direction ψ ∈ X is defined as
if the limit exists. If the limit exists for all ψ ∈ X, then one says that F is Gâteaux differentiable at u.
The limit appearing in is taken relative to the topology of Y. If X and Y are real topological vector spaces, then the limit is taken for real τ. On the other hand, if X and Y are complex topological vector spaces, then the limit above is usually taken as τ→0 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 Gâteaux derivative.
This function is homogeneous in the sense that for all scalars α
However, this function need not be additive, so that the Gâteaux differential may fail to be linear, unlike the Fréchet derivative
. Even if linear, it may fail to depend continuously on ψ if X and Y are infinite dimensional. Furthermore, for Gâteaux differentials that are linear and continuous in ψ, there are several inequivalent ways to formulate their continuous differentiability.
For example, consider the real-valued function F of two real variables defined by
This is Gâteaux differentiable at (0, 0), with its differential there being
However this is continuous but not linear in the arguments (a,b). In infinite dimensions, any discontinuous linear functional on X is Gâteaux differentiable, but its Gâteaux differential at 0 is linear but not continuous.
Relation with the Fréchet derivative
If F is Fréchet differentiable, then it is also Gâteaux differentiable, and its Fréchet and Gâteaux derivatives agree. The converse is clearly not true, since the Gâteaux derivative may fail to be linear or continuous. In fact, it is even possible for the Gâteaux derivative to be linear and continuous but for the Fréchet derivative to fail to exist.
Nevertheless, for functions F from a complex Banach space X to another complex Banach space Y, the Gâteaux derivative (where the limit is taken over complex τ tending to zero as in the definition of complex differentiability
) is automatically linear, a theorem of . Furthermore, if F is (complex) Gâteaux differentiable at each u ∈ U with derivative
then F is Fréchet differentiable on U with Fréchet derivative DF . 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 Gâteaux differentiability may be defined in two inequivalent ways. Suppose that F:U→Y is Gâteaux differentiable at each point of the open set U. One notion of continuous differentiability in U requires that the mapping on the product space
be continuous. Linearity need not be assumed: if X and Y are Fréchet spaces, then dF(u;•) is automatically bounded and linear for all u .
A stronger notion of continuous differentiability requires that
be a continuous mapping
from U to the space of continuous linear functions from X to Y. Note that this already presupposes the linearity of DF(u).
As a matter of technical convenience, this latter notion of continuous differentiability is typical (but not universal) when the spaces X and Y are Banach, since L(X,Y) 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 function
s on a manifold
.
Rather than a multilinear function, this is instead a homogeneous function
of degree n in h.
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 F, at least in the special case where F is scalar-valued. However, this may fail to have any reasonable properties at all, aside from being separately homogeneous in h and k. It is desirable to have sufficient conditions in place to ensure that D2F(u){h,k} is a symmetric bilinear function of h and k, and that it agrees with the polarization
of dnF.
For instance, the following sufficient condition holds . Suppose that F is C1 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 D2F(u){h,k} is bilinear and symmetric in h and k. By virtue of the bilinearity, the polarization identity holds
relating the second order derivative D2F(u) with the differential d2F(u;−). Similar conclusions hold for higher order derivatives.
holds for the Gâteaux derivative of F, provided F 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:
be the Hilbert space
of square-integrable function
s on a Lebesgue measurable set
in the Euclidean space
RN. The functional
given by
where F is a real
-valued function of a real variable with F′ = ƒ and u is defined on Ω with real values, has Gâteaux derivative
Indeed,
Letting τ → 0 gives the Gâteaux derivative
that is, the inner product 〈ƒ,ψ〉.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Gâteaux differential or Gâteaux derivative is a generalization of the concept of directional derivative
Directional derivative
In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P in the direction of V...
in differential calculus
Differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....
. Named after René Gâteaux
René Gâteaux
René Eugène Gateaux , was a French mathematician. He is known for the Gâteaux derivative. Part of his work has been posthumously published by Paul Lévy. Gâteaux was killed during World War I.-External links:...
, a French mathematician who died young in World War I
World War I
World War I , which was predominantly called the World War or the Great War from its occurrence until 1939, and the First World War or World War I thereafter, was a major war centred in Europe that began on 28 July 1914 and lasted until 11 November 1918...
, it is defined for functions between locally convex topological vector space
Topological vector space
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
s such as Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s. Like the Fréchet derivative
Fréchet derivative
In mathematics, the Fréchet derivative is a derivative defined on Banach spaces. Named after Maurice Fréchet, it is commonly used to formalize the concept of the functional derivative used widely in the calculus of variations. Intuitively, it generalizes the idea of linear approximation from...
on a Banach space, the Gâteaux differential is often used to formalize the functional derivative
Functional derivative
In mathematics and theoretical physics, the functional derivative is a generalization of the gradient. While the latter differentiates with respect to a vector with discrete components, the former differentiates with respect to a continuous function. Both of these can be viewed as extensions of...
commonly used in the calculus of variations
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
and physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
.
Unlike other forms of derivatives, the Gâteaux differential of a function may be nonlinear. However, often the definition of the Gâteaux differential also requires that it be a continuous linear transformation. Some authors, such as , draw a further distinction between the Gâteaux differential (which may be nonlinear) and the Gâteaux 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
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
in the context of infinite dimensional holomorphy or continuous differentiability in nonlinear analysis.
Definition
Suppose X and Y are locally convex topological vector spaceTopological vector space
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
s (for example, Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s), U ⊂ X is open, and F : X → Y. The Gâteaux differential dF(u;ψ) of F at u ∈ U in the direction ψ ∈ X is defined as
if the limit exists. If the limit exists for all ψ ∈ X, then one says that F is Gâteaux differentiable at u.
The limit appearing in is taken relative to the topology of Y. If X and Y are real topological vector spaces, then the limit is taken for real τ. On the other hand, if X and Y are complex topological vector spaces, then the limit above is usually taken as τ→0 in the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
as in the definition of complex differentiability
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
. In some cases, a weak limit
Weak topology
In mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual...
is taken instead of a strong limit, which leads to the notion of a weak Gâteaux derivative.
Linearity and continuity
At each point u ∈ U, the Gâteaux differential defines a functionThis function is homogeneous in the sense that for all scalars α
However, this function need not be additive, so that the Gâteaux differential may fail to be linear, unlike the Fréchet derivative
Fréchet derivative
In mathematics, the Fréchet derivative is a derivative defined on Banach spaces. Named after Maurice Fréchet, it is commonly used to formalize the concept of the functional derivative used widely in the calculus of variations. Intuitively, it generalizes the idea of linear approximation from...
. Even if linear, it may fail to depend continuously on ψ if X and Y are infinite dimensional. Furthermore, for Gâteaux differentials that are linear and continuous in ψ, there are several inequivalent ways to formulate their continuous differentiability.
For example, consider the real-valued function F of two real variables defined by
This is Gâteaux differentiable at (0, 0), with its differential there being
However this is continuous but not linear in the arguments (a,b). In infinite dimensions, any discontinuous linear functional on X is Gâteaux differentiable, but its Gâteaux differential at 0 is linear but not continuous.
Relation with the Fréchet derivative
If F is Fréchet differentiable, then it is also Gâteaux differentiable, and its Fréchet and Gâteaux derivatives agree. The converse is clearly not true, since the Gâteaux derivative may fail to be linear or continuous. In fact, it is even possible for the Gâteaux derivative to be linear and continuous but for the Fréchet derivative to fail to exist.
Nevertheless, for functions F from a complex Banach space X to another complex Banach space Y, the Gâteaux derivative (where the limit is taken over complex τ tending to zero as in the definition of complex differentiability
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
) is automatically linear, a theorem of . Furthermore, if F is (complex) Gâteaux differentiable at each u ∈ U with derivative
then F is Fréchet differentiable on U with Fréchet derivative DF . This is analogous to the result from basic complex analysis
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
that a function is analytic
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
if it is complex differentiable in an open set, and is a fundamental result in the study of infinite dimensional holomorphy.
Continuous differentiability
Continuous Gâteaux differentiability may be defined in two inequivalent ways. Suppose that F:U→Y is Gâteaux differentiable at each point of the open set U. One notion of continuous differentiability in U requires that the mapping on the product space
be continuous. Linearity need not be assumed: if X and Y are Fréchet spaces, then dF(u;•) is automatically bounded and linear for all u .
A stronger notion of continuous differentiability requires that
be a continuous mapping
from U to the space of continuous linear functions from X to Y. Note that this already presupposes the linearity of DF(u).
As a matter of technical convenience, this latter notion of continuous differentiability is typical (but not universal) when the spaces X and Y are Banach, since L(X,Y) 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
Differentiation in Fréchet spaces
In mathematics, in particular in functional analysis and nonlinear analysis, it is possible to define the derivative of a function between two Fréchet spaces. This notion of differentiation is significantly weaker than the derivative in a Banach space. Nevertheless, it is the weakest notion of...
has applications such as the Nash–Moser inverse function theorem in which the function spaces of interest often consist of smooth function
Smooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
s on a manifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
.
Higher derivatives
Whereas higher order Fréchet derivatives are naturally defined as multilinear functions by iteration, using the isomorphisms Ln(X,Y) = L(X, Ln−1(X,Y)), higher order Gâteaux derivative cannot be defined in this way. Instead the nth order Gâteaux derivative of a function F : U⊂X → Y in the direction h is defined byRather than a multilinear function, this is instead a homogeneous function
Homogeneous function
In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then...
of degree n in h.
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 F, at least in the special case where F is scalar-valued. However, this may fail to have any reasonable properties at all, aside from being separately homogeneous in h and k. It is desirable to have sufficient conditions in place to ensure that D2F(u){h,k} is a symmetric bilinear function of h and k, and that it agrees with the polarization
Polarization of an algebraic form
In mathematics, in particular in algebra, polarization is a technique for expressing a homogeneous polynomial in a simpler fashion by adjoining more variables...
of dnF.
For instance, the following sufficient condition holds . Suppose that F is C1 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 D2F(u){h,k} is bilinear and symmetric in h and k. By virtue of the bilinearity, the polarization identity holds
relating the second order derivative D2F(u) with the differential d2F(u;−). Similar conclusions hold for higher order derivatives.
Properties
A version of the fundamental theorem of calculusFundamental theorem of calculus
The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...
holds for the Gâteaux derivative of F, provided F is assumed to be sufficiently continuously differentiable. Specifically:
- Suppose that F : X → Y is C1 in the sense that the Gâteaux derivative is a continuous function dF : U×X→Y. Then for any u∈U and h∈X,
-
- 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 u ∈ U and x ∈ X.
- (Taylor's theorem with remainder.)
-
- Suppose that the line segment between u ∈ U and u+h lies entirely within U. If F is Ck then
- where the remainder term is given by
Example
Let be the Hilbert spaceHilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
of square-integrable function
Square-integrable function
In mathematics, a quadratically integrable function, also called a square-integrable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite...
s on a Lebesgue measurable set
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
in the Euclidean spaceEuclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
RN. The functional
given by
where F is a real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
-valued function of a real variable with F′ = ƒ and u is defined on Ω with real values, has Gâteaux derivative
Indeed,
Letting τ → 0 gives the Gâteaux derivative
that is, the inner product 〈ƒ,ψ〉.
See also
- Derivative (generalizations)Derivative (generalizations)The derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, and geometry.- Derivatives in analysis :...
- Differentiation in Fréchet spacesDifferentiation in Fréchet spacesIn mathematics, in particular in functional analysis and nonlinear analysis, it is possible to define the derivative of a function between two Fréchet spaces. This notion of differentiation is significantly weaker than the derivative in a Banach space. Nevertheless, it is the weakest notion of...
- Quasi-derivativeQuasi-derivativeIn mathematics, the quasi-derivative is one of several generalizations of the derivative of a function between two Banach spaces. The quasi-derivative is a slightly stronger version of the Gâteaux derivative, though weaker than the Fréchet derivative....