Part of a series of articles about 
Calculus 

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 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 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 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. Furthermore, for Gateaux differentials that are linear and continuous in there are several inequivalent ways to formulate their continuous differentiability.
For example, consider the realvalued function of two real variables defined by
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
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
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 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 scalarvalued. 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
A version of the fundamental theorem of calculus holds for the Gateaux derivative of provided 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 higherorder derivatives. Further properties, also consequences of the fundamental theorem, include:
Let be the Hilbert space of squareintegrable functions on a Lebesgue measurable set in the Euclidean space The functional
Indeed, the above is the limit of