Gâteaux derivative

The Gâteaux differential, named after René Gâteaux (1889-1914) represents a generalization of the ordinary Differentiationsbegriffes by defining the directional derivative in infinite-dimensional spaces. Generally has a function for open amount is differentiable at the point, as a definition of the partial derivative

In particular, results for the known differential

The Gâteaux differential generalized these concepts to infinite-dimensional vector spaces.

Definitions

Weierstraß decomposition formula

With open and normed spaces. Then say in Gateaux differentiable if there exists a linear function, so that

For all with. This is equivalent to:

Then is called the Gateaux derivative of the point.

First variation; variation derivation

Let now given the following situation for the Gâteaux differential: It was, as usual, into defined functional; is a normed linear space ( that is, a vector space equipped with a norm) or a more general topological vector space with conditions about which one has to make further thoughts in a specific application; further and was. Then Gâteux differential at the point in the direction, if it exists, is defined by the following derivation by:

Or also by

It should be noted here, and also in it, but.

Gâteux the derivative with respect to with respect to the size of a functional, which is also referred to as a first variation of the position.

Another possibility is to use instead of normed vector spaces more general topological vector spaces with a corresponding notion of convergence. Especially in physics books functionals are usually denoted by the letter, and instead of the size to write mostly with distributionswertigen sizes. Instead of the derivation is carried out in an additional step, a variation of the so-called derivation, which is closely related to the Gâteaux derivative.

Example

Obtained after a partial integration with vanishing ausintegrierten part a result of the shape with the variation of derivative

( The variation of discharge " of the point Q (t) " for continuous variables is thus a generalization of the partial derivative of a function of n variables, thus for example, for the nominal case. Thus, similar to the imaginary case, the total differential of a function of n variables so also here, the total differential of the functional invariant meaning. further details in Chapter Lagrangian formalism. )

In the following, waived by the " boldface " letters because of the simplicity on the identification of vectors.

Second variation

Half -page differential and directional derivative

Under the same conditions as above the unilateral Gâteaux differential is by

Or by

Defined. The unilateral Gâteaux differential is also called differential direction of at the site. Namely generalized for the vector associated with the direction of " continuous variables " unilateral Gâteaux differential ( more precisely, the corresponding variation derivative) just the directional derivative of in direction at the site.

Gâteaux - derivative

Is in constant, linear functional (ie, mediated by the function is homogeneous, additive and steadily in the argument ), then that means Gâteaux - derivative at. and Gâteaux - differentiable at.

Properties of the first variation

• The Gâteaux differential is homogeneous, which means for everyone. The property also applies to the unilateral Gâteaux differential.

• Gâteux the differential is not linear. In general case, therefore, applies For example, that the differential is Gâteux nonlinear look for and, wherein, then. The function is not linear. It applies, for example.

Examples

(where )

Applications

Such as the usual discharge is Gâteux differential for determining extremes and thus in the optimization of use. Be open, normed linear space ( the interior of the lot ), and the open ball around with radius. Necessary optimality condition: Be a local minimum of on, then, if the one-sided Gâteaux differential exists in. Sufficient optimality condition: possess in a second variation and. Case may be, and for one and then is strict local minimizer of on.