Gâteaux derivative

Formally, the definition of differentiation is identical to the Gâteaux derivative.

The converse is clearly not true, since the Gâteaux derivative may fail to be linear or continuous.

The directional derivative is a special case of the Gâteaux derivative.

He is known for the Gâteaux derivative.

See the articles on the Fréchet derivative and the Gâteaux derivative.

The Gâteaux derivative extends the concept to locally convex topological vector spaces.

For counterexamples, see Gâteaux derivative.

In some cases, a weak limit is taken instead of a strong limit, which leads to the notion of a weak Gâteaux derivative.

The quasi-derivative is a slightly stronger version of the Gâteaux derivative, though weaker than the Fréchet derivative.

(Here φ denotes the Gâteaux derivative of φ.)

The Gâteaux derivative allows for an extension of a directional derivative to locally convex topological vector spaces.

The Fréchet derivative should be contrasted to the more general Gâteaux derivative which is a generalization of the classical directional derivative.

In the latter case the search space is typically a function space, and one calculates the Gâteaux derivative of the functional to be minimized to determine the descent direction.

In this case, the iterated Gâteaux derivatives are multilinear in the h, but will in general fail to be continuous when regarded over the whole space X.

There is a generalization both of the directional derivative, called the Gâteaux derivative, and of the differential, called the Fréchet derivative.

Finally, if f is quasi-differentiable, then it is Gâteaux differentiable and its Gâteaux derivative is equal to its quasi-derivative.

In mathematics, the Gâteaux differential or Gâteaux derivative is a generalization of the concept of directional derivative in differential calculus.

A version of the fundamental theorem of calculus holds for the Gâteaux derivative of F, provided F is assumed to be sufficiently continuously differentiable.

For instance, the Fréchet or Gâteaux derivative can be used to define a notion of a holomorphic function on a Banach space over the field of complex numbers.

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).

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

This approach allows the differential (as a linear map) to be developed for a variety of more sophisticated spaces, ultimately giving rise to such notions as the Fréchet or Gâteaux derivative.

These cases can occur because the definition of the Gâteaux derivative only requires that the difference quotients converge along each direction individually, without making requirements about the rates of convergence for different directions.

For example, when the space of functions is a Banach space, the functional derivative becomes known as the Fréchet derivative, while one uses the Gâteaux derivative on more general locally convex spaces.

This is a major difference between the theory of Banach spaces and that of Fréchet spaces and necessitates a different definition for continuous differentiability of functions defined on Fréchet spaces, the Gâteaux derivative: