Functional (mathematics)

The functional referred to in mathematics mostly a function of a vector space V in the field K over which the vector space is modeled.

Often V is a function space, ie a vector space whose elements are real or complex-valued functions. A functional is thus a function of functions. The mathematical part of the field of functional analysis got its name because it emerged historically from the study of such functionals.

As a fundamental distinction, it is useful to consider separately the linear and nonlinear functionals, since these two types of functionals are treated in very different ways in mathematics. In both cases, we restrict ourselves to the almost exclusively examined cases in which the number field ( Skalarkörper ) is the field of real numbers or the field of complex numbers.

• 4.1 Examples of the non-linear functional

Definition

Be a vector space with. A functional is a mapping

Simple examples

A simple linear functional on the vector space of functions on the real axis is the evaluation functional at zero, the so-called Dirac delta:

A simple non- linear functional on the vector space of curves in space, especially continuously differentiable functions from here after, is the arc length functional:

Linear Functional

In most areas of functional analysis, such as in the theory of topological vector spaces, the term functional is used ( without further addition ) as a synonym for linear functionals. Such a functional is defined as a linear transformation of the vector space in its Skalarkörper. The amount of all these functionals is again in its natural form is a vector space over the same body, by defining two functionals and the addition and scalar multiplication pointwise, ie

The vector space of linear functionals on the vector space is called the algebraic dual space and often referred to.

Examples of dual spaces

For the vector space of the dual space is canonically isomorphic to the vector space itself, ie. The canonical isomorphism is thereby mediated via the standard scalar product:

The same is true for the vector space as in the first case, however, the canonical map in this case semilinear:

The dual space is in this case equal in size, but has respect to the canonical figure another scalar multiplication. In terms of linear algebra, one also says the dual space is canonically isomorphic to the complex conjugate vector space.

For general finite-dimensional vector spaces can be shown by choosing a basis and application of the first two cases, that the dual space always has the same dimension as the source space. The maps between the vector space and the dual space are then but generally not canonical.

The case is much more complicated for infinite-dimensional vector spaces. In some important cases, such as for Hilbert spaces, the vector space is indeed a canonical subspace, in general, however, this does not apply. The algebraic dual space of an infinite-dimensional vector space has also increasingly important dimension (in the sense of cardinality an algebraic basis) as the origin of space.

Continuous linear functionals

As just seen, the algebraic dual space of an infinite-dimensional vector space is always greater than or equal to the original vector space. One can even argue that these dual spaces are often huge and contain many elements that are mathematically hardly manageable. The goal of functional analysis it is, however, to extend the methods of multidimensional analysis on infinite-dimensional spaces and to investigate particular concepts such as continuity and differentiability. Therefore, a priori, be considered only vector spaces, at least allow the concept of continuity makes sense. These are topological vector spaces, to which all normed vector spaces, in particular Banach spaces and Hilbert spaces belong.

In a topological vector space all linear functionals are now generally not continuous. The amount of the continuous functional, that is in the functional analysis of primary interest, is the topological and dual space is designated.

Examples of topological dual spaces

For finite dimensional vector spaces there is a natural topology ( norm topology ), which arises from the Euclidean norm (specifically from an arbitrary Euclidean norm, if one chooses a basis). This is just the topology, which is the normal default Analysis based, and in that each linear functional is continuous. That is, the algebraic dual space is equal to the topological dual space.

In the infinite-dimensional case, the topological dual space is (almost) always a proper subspace of the algebraic dual space.

In normed vector spaces, a functional is continuous if it is limited, that is,

The topological dual space is then automatically a Banach space with the supremum norm defined above.

In Hilbert spaces is the topological dual space canonically with the origination space identified ( Rieszscher representation theorem ). The identification is done as in the finite case, the scalar product:

The topological dual space of the space of infinitely many times continuously differentiable functions with compact support on the real axis ( the so-called test functions ) with a certain ( explained here in detail ) topology is called the space of distributions. In this room there is also the above -mentioned example of the Dirac delta functional.

Nonlinear Functional

Nonlinear Functional occurred historically the first time in Calculus of Variations. Your study is fundamentally different from that of the linear functionals described above. In the calculus of variations you put it, for example, aims to determine the extreme points of such functional points. For this purpose, one needs a generalization of the concept of deriving the multi-dimensional analysis, i.e., a definition of the differential of the functional. In the calculus of variations and in the applications of this differential is known as the variation of dissipation, the term is more precise mathematically, for example, by the Fréchet derivative and the Gateaux derivative.

Examples of non-linear functional

Of great importance in the application, especially in classical mechanics have non-linear functionals on spaces curve, as in the example of the arc length functionals above. One can easily generalize this example.

We again consider a curve space and additionally a continuously differentiable function. Thus we define:

It is said that the functional L has a stationary point at a curve c, when the differential

For all variations h, the curves are the beginning and end in the zero disappears. This here is exactly the case when the (ordinary ) differential of F disappears around the curve c:

Considering a curve space and doubly continuous functions with two arguments, we obtain analogously:

Stationary points on a curve C, when the differential

For all variations h, disappears. This is in this simple case, exactly the case when c satisfy the Euler-Lagrange equation, i.e.

Sometimes, especially in application-oriented text, you write a functional dependency (as opposed to the usual functional dependence) with square or curly instead of parentheses referring, possibly a dummy argument of the function argument, so I [ f] or I { f ( x) } instead of I (f).