This is just the usual Fréchet derivative - an example of a differential 1-form.

See, for instance, the Fréchet derivative article.

The Fréchet derivative in finite-dimensional spaces is the usual derivative.

In the infinite dimensional case it is required that the Fréchet derivative have a bounded inverse at p.

For an example of the use of this test on a Banach space, see the article Fréchet derivative.

A further generalization for a function from one Banach space to another is the Fréchet derivative.

The chain rule is also valid for Fréchet derivatives in Banach spaces.

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

One needs the Fréchet derivative to be boundedly invertible at each in order for the method to be applicable.

Suppose that is a complex-valued function which is Fréchet derivative as a function .

Even if you assume that X is a Banach space and ask whether a Fréchet derivative exists almost everywhere, this does not hold.

In mathematics, the Fréchet derivative is a derivative defined on Banach spaces.

In the above, Dφ(x) denotes the Fréchet derivative of φ at x.

The Fréchet derivative allows the extension of the directional derivative to a general Banach space.

However, this function need not be additive, so that the Gâteaux differential may fail to be linear, unlike the Fréchet derivative.

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

In fact, it is even possible for the Gâteaux derivative to be linear and continuous but for the Fréchet derivative to fail to exist.

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

If a sequence of points is chosen along these directions, the quotient in the definition of the Fréchet derivative, which considers all directions at once, may not converge.

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

Thus, in order for a linear Gâteaux derivative to imply the existence of the Fréchet derivative, the difference quotients have to converge uniformly for all directions.

In mathematics, Kachurovskii's theorem is a theorem relating the convexity of a function on a Banach space to the monotonicity of its Fréchet derivative.

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.

The Fréchet derivative has applications to nonlinear problems throughout mathematical analysis and physical sciences, particularly to the calculus of variations and much of nonlinear analysis and nonlinear functional analysis.

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.