Gâteaux derivative

In mathematics, the Gâteaux differential is a generalisation 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 spaces such as Banach spaces. 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. If the Gâteaux differential is linear and continuous, then the resulting linear operator is called the Gâteaux derivative.

Definition

Suppose $X$ and $Y$ are locally convex topological vector spaces (for example, Banach spaces), $Usubset\; X$ is open, and

:$F:X\; ightarrow\; Y.$

The Gâteaux differential $dF(u,psi)$ of $F$ at $uin\; U$ in the direction $psiin\; X$ is defined as

:$dF(u,psi)=lim\_\{\; au\; ightarrow\; 0\}frac\{F(u+\; au\; psi)-F(u)\}\{\; au\}=left.frac\{d\}\{d\; au\}F(u+\; au\; psi)\; ight|\_\{\; au=0\}$

if the limit exists. If the limit exists for all $psi\; in\; X$, then one says that $F$ has Gâteaux differential at $uin\; U$. If "X" and "Y" are complex topological vector spaces, then the limit above is usually taken as τ→0 in the complex plane. If, on the other hand, "X" and "Y" are real, then the limit is taken for real τ.

The Gâteaux differential may fail to be linear or, being linear, it may fail to be a continuous linear transformation. If the mapping

:$DF(u)\; :\; psimapsto\; dF(u,psi)$

is continuous and linear, then it is called the Gâteaux derivative of "F" at "u" and "F" is said to be Gâteaux differentiable at "u".

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

:$dF:U\; imes\; X\; ightarrow\; Y\; ,$

be continuous. Another notion requires continuity of the mapping

:$umapsto\; DF(u)\; ,$

be a continuous mapping

:$U\; o\; L(X,Y)\; ,$

from "U" to the space of continuous linear functions from "X" to "Y". The latter notion of continuous differentiability is typical (but not universal) when the spaces "X" and "Y" are Banach. The former is the more common definition in applications such as the Nash-Moser inverse function theorem.

Properties

If the Gâteaux differential exists, it is unique.

For each $uin\; U$ the Gâteaux differential is an operator

:$dF(u,cdot):X\; ightarrow\; Y.$

This operator is homogeneous, so that

:$dF(u,alphapsi)=alpha\; dF(u,psi),$,

but it is not additive in general case, and, hence, is not always linear, unlike the Fréchet derivative.

However, suppose that "X" and "Y" complex Banach spaces, "u" is a point of the open set "U"⊂"X", and "F":"U" → "Y". Then if "F" is (complex) Gâteaux differentiable at each "u" ∈ "U" with derivative

:$DF(u)\; :\; psimapsto\; dF(u,psi)$

then "F" is Fréchet differentiable on "U" with Fréchet derivative "DF" harv|Zorn|1946. This is analogous to the result from basic complex analysis that a function is analytic if it is complex differentiable, and is a fundamental result in the study of infinite dimensional holomorphy.

If "F" is Fréchet differentiable, then it is also Gâteaux differentiable, and its Fréchet and Gâteaux derivatives agree.

Example

Let $X$ be the Hilbert space of square-integrable functions on a Lebesgue measurable set $Omega$ in the Euclidean space R^"N". The functional

:$E:X\; ightarrow\; mathbb\{R\}$

given by

:$E(u)=int\_Omega\; Fleft(u(x)\; ight)dx$

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 :$dE(u,psi)=langle\; f(u),psi\; angle,.$

Indeed,

:$frac\{E(u+\; aupsi)\; -\; E(u)\}\{\; au\}\; =\; frac\{1\}\{\; au\}\; left(\; int\_Omega\; F(u+\; aupsi)dx\; -\; int\_Omega\; F(u)dx\; ight)$:$quadquad\; =frac\{1\}\{\; au\}\; left(\; int\_Omegaint\_0^1\; frac\{d\}\{ds\}\; F(u+s\; aupsi)\; ,ds,dx\; ight)$:$quad\; quad\; =int\_Omegaint\_0^1\; f(u+s\; aupsi)psi\; ,ds,dx.$

Letting τ → 0 gives the Gâteaux derivative:$dE(u,psi)\; =\; int\_Omega\; f(u(x))psi(x)\; ,dx,$that is, the inner product 〈ƒ,ψ〉.

See also

* Derivative (generalizations)
* Differentiation in Fréchet spaces

References

* citation | first = R|last=Gâteaux | title =Sur les fonctionnelles continues et les fonctionnelles analytiques | pages = | url = http://gallica.bnf.fr/ | journal = Comptes rendus de l'academie des sciences|publication-place=Paris|volume=157|year=1913 | pages = 325-327 | accessmonthday=30 July |accessyear = 2006 .
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*|doi=10.1090/S0002-9904-1946-08524-9.