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.
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.
are complex topological vector spaces, then the limit above is usually taken as
In some cases, a weak limit is taken instead of a strong limit, which leads to the notion of a weak Gateaux derivative.
However, this function need not be additive, so that the Gateaux differential may fail to be linear, unlike the Fréchet derivative.
there are several inequivalent ways to formulate their continuous differentiability.
In infinite dimensions, any discontinuous linear functional on
The converse is clearly not true, since the Gateaux derivative may fail to be linear or continuous.
tending to zero as in the definition of complex differentiability) is automatically linear, a theorem of Zorn (1945).
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.
is Gateaux differentiable at each point of the open set
A stronger notion of continuous differentiability requires that
As a matter of technical convenience, this latter notion of continuous differentiability is typical (but not universal) when the spaces
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.
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.
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
However, this may fail to have any reasonable properties at all, aside from being separately homogeneous in
It is desirable to have sufficient conditions in place to ensure that
For instance, the following sufficient condition holds (Hamilton 1982).
By virtue of the bilinearity, the polarization identity holds
Similar conclusions hold for higher order derivatives.
A version of the fundamental theorem of calculus holds for the Gateaux derivative of
Further properties, also consequences of the fundamental theorem, include: Let
be the Hilbert space of square-integrable functions on a Lebesgue measurable set