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In mathematics, the **Gateaux differential** or **Gateaux derivative** is a generalization of the concept of directional derivative in differential calculus. Named after René Gateaux, 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 a nonlinear operator. 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.

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

(1) |

If the limit exists for all then one says that is Gateaux differentiable at

The limit appearing in (**1**) 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.

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 (i.e. in the case that is an unbounded linear operator). 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 (where the limit is taken over complex tending to zero as in the definition of complex differentiability) is automatically linear, a theorem of Zorn (1945). Furthermore, if is (complex) Gateaux differentiable at each with derivative
then is Fréchet differentiable on with Fréchet derivative (Zorn 1946). 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 (Hamilton 1982).

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.

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

(2) |

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

(3) |

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 (Hamilton 1982). Suppose that is in 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. 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.

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) (Vainberg (1964)).

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*not*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

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

- Hadamard derivative
- Derivative (generalizations) – Fundamental construction of differential calculus
- Differentiable vector-valued functions from Euclidean space – Differentiable function in functional analysis
- Differentiation in Fréchet spaces
- Fractal derivative – Generalization of derivative to fractals
- Generalizations of the derivative – Fundamental construction of differential calculus
- Infinite-dimensional vector function – function whose values lie in an infinite-dimensional vector space
- Quasi-derivative – Generalization of a derivative of a function between two Banach spaces
- Quaternionic analysis – Function theory with quaternion variable
- Semi-differentiability

- Gateaux, René (1913), "Sur les fonctionnelles continues et les fonctionnelles analytiques",
*Comptes rendus hebdomadaires des séances de l'Académie des sciences*,**157**, Paris: 325–327, retrieved 2 September 2012. - Gateaux, René (1919), "Fonctions d'une infinité de variables indépendantes",
*Bulletin de la Société Mathématique de France*,**47**: 70–96, doi:10.24033/bsmf.995. - Hamilton, R. S. (1982), "The inverse function theorem of Nash and Moser",
*Bull. Amer. Math. Soc.*,**7**(1): 65–222, doi:10.1090/S0273-0979-1982-15004-2, MR 0656198 - Hille, Einar; Phillips, Ralph S. (1974),
*Functional analysis and semi-groups*, Providence, R.I.: American Mathematical Society, MR 0423094. - Tikhomirov, V.M. (2001) [1994], "Gâteaux variation",
*Encyclopedia of Mathematics*, EMS Press. - Vainberg, M.M. (1964),
*Variational Methods for the Study of Nonlinear Operators*, San Francisco, London, Amsterdam: Holden-Day, Inc, p. 57 - Zorn, Max (1945), "Characterization of analytic functions in Banach spaces",
*Annals of Mathematics*, Second Series,**46**(4): 585–593, doi:10.2307/1969198, ISSN 0003-486X, JSTOR 1969198, MR 0014190. - Zorn, Max (1946), "Derivatives and Frechet differentials",
*Bulletin of the American Mathematical Society*,**52**(2): 133–137, doi:10.1090/S0002-9904-1946-08524-9, MR 0014595.