Function space

In mathematics, a function space is a set of functions that all have the same domain. However, as the mathematical concept of space are not sharply defined, the term functional space similar.

Usually a function room with a vector addition and scalar multiplication is provided, so that it forms a vector space, then it is called a linear function space. Many important linear function spaces are infinite dimensional. These form an important object of study in functional analysis. Linear function spaces are often provided with a standard, so that a normed space or - in the case of completeness - even a Banach space is created. In other cases, linear function spaces by defining a topology on a topological vector space or a locally convex space.

• 3.1 topology
• 3.2 Functional Analysis

Terminology

Function rooms are in the field of linear algebra, vector spaces, whose elements are understood as functions. Mainly function spaces are, however, considered in the field of functional analysis. Herein is meant having a topological structure with a working space is a vector space, the elements are regarded as features.

In linear algebra

Let be a non-empty set and a vector space over a field, then (also referred to or ) the set of all functions from to. The amount is for and for scalars by the

• Addition and
• The scalar multiplication

To a vector space. This vector space and the subspaces of are referred to in the field of linear algebra as a linear function space.

In topology

In topology is meant by a functional space a topological space whose elements are functions of a set or a topological space in a topological space and its topology derived from the topology of and and any additional structures, such as a metric or a uniform structure is.

In the functional analysis

Let be a non-empty set, a topological vector space (often a Banach space or locally convex vector space ) and the vector space of all functions from to. A linear function space in the area of functional analysis is a subspace of which is provided with a topological structure of derived.

History

The history of the function rooms can be divided into three phases. The first phase began around the beginning of the twentieth century and lasted until the mid -1930s. In this time the function rooms of the times continuously - differentiable functions, as well as the classical Lebesgue spaces of integrable functions. In addition, the spaces of Hölder continuous functions and the classical Hardy spaces are still expected at this stage.

The second, the constructive phase began with the publications by Sergei Sobolev Lvovitch from the years 1935 to 1938, where he introduced the now named after him ( integer ) Sobolev spaces. The theory of distributions arose and new techniques, such as embedding theorems have been developed for solving partial differential equations. In this phase function rooms are equipped with norms or quasi- norms. Important newly developed spaces for this period are the Zygmund spaces ( or classes) that Slobodeckij areas, the classical Besov spaces and the Bessel potential spaces. Also the BMO space of John Fritz and Louis Nirenberg and the real Hardy spaces of Elias Stein and Guido Weiss were introduced in the 1960s.

The third phase, which is referred to as systematic phase began in the 1960s and coincided clearly with the design phase. Here the techniques of Fourier analysis were developed further and called Maximalungleichungen investigated. Using these tools, the Besov - Lebesgue spaces and Lizorkin - Triebel spaces which have been developed. These two rooms can be embedded into the space of tempered distributions. How suggest their definitions, these spaces are very closely intertwined with Fourier Analysis.

Examples

Topology

• Are and topological spaces, we write for the set of continuous functions.
• Is given to a metric, then one can speak meaningfully of the amount of bounded functions ( without topology ). For this figure amount, among other things, the notation will be used. Is also defined in a topology, to write to the limited amount of continuous functions. In these spaces is

• Are on the topology and given by a pseudo- metric or a metric, then to write the set of continuous functions uniformly. Are and uniform spaces, then this notation denotes the set of uniform continuous functions, ie those functions which respect the uniform structures.
• Is the field of real numbers or complex numbers and is clear from the context in which the body reflect the functions, it is usually omitted from the notation, and then to write short, respectively.

Functional Analysis

Most functional spaces are investigated in functional analysis. The following list is a selection of the investigated there rooms. Be the definition of quantity of the studied functions. Then

• The space of - times continuously differentiable functions with. If it is compact, the space with respect to the usual standard

• The space of times continuously differentiable functions Holder continuous with the exponent. Is compact, it is provided with the standard

• The space of test functions. It contains all smooth functions with compact support and is equipped with the topology induced by the notion of convergence. A sequence converges to if there is a compact set with for all j, and

• The space of Lebesgue integrable functions (see Lp). This space does not consist of individual features, but of equivalence classes of functions which differ only by a Lebesgue -null set. For this reason, the standard also

• The space of locally integrable functions. Let be a measurable function. Locally integrable means that for all compact subsets of the integral

• The space of weakly differentiable functions. He bears the name of Sobolev space. This space is often used as an ansatz space for solving differential equations. For any continuously differentiable function is also weakly differentiable.

• The space of functions which fall faster than any polynomial function. The set is called the Schwartz space, named after the, French mathematician Laurent Schwartz. The space has been constructed so that the Fourier transform is an isomorphism on it. The dual space of the Schwartz space is the space of tempered distributions.

• By perceives a real or complex number sequences as pictures of or after, you can understand every vector space of sequences as a function room.

• Is the space of holomorphic functions. These functions are infinitely differentiable, and its Taylor series converges to the output function. Often called holomorphic functions analytically. Sometimes you listed this space with too.

• Is the space of holomorphic, integrable functions, it is called Hardy space and is an analogue of the space. Usually used as a set of definitions, the unit sphere.

Functional areas in theoretical computer science

This particular function spaces are used in conjunction with models of the lambda calculus. Its objects occur equally as functions, but also as their arguments and results. It is therefore desirable a subject area, the function space is isomorphic to itself, which is not possible from Kardinalitätsgründen but. Dana Scott in 1969 could solve this problem by restricting to continuous functions with respect to a suitable topology. Refers to the continuous functions of a complete partial order, then. This form of function spaces is now the subject of field theory. Later a also suitable function space are found as retraction of an object in a cartesian closed category.