Function room

In mathematics , a function space is a set of functions that all have the same domain . However, like the mathematical concept of space , the term function space cannot be sharply delimited.

Usually a function space is provided with vector addition and scalar multiplication , so that it forms a vector space , then one speaks of a linear function space . Many important linear function spaces are infinitely dimensional . These form an important subject of investigation in functional analysis . Linear function spaces are often given a norm so that a normalized space or - in the case of completeness - even a Banach space is created. In other cases, by defining a topology , linear function spaces become a topological vector space or a locally convex space .

Terminology

In the area of linear algebra , function spaces are vector spaces , the elements of which are understood as functions . Functional spaces are mainly considered in the area of functional analysis. Here, a function space is understood to be a vector space with a topological structure, the elements of which are understood as functions.

In linear algebra

Addition in the function space: The sum of the sine function and the exponential function is with ${\displaystyle \sin +\exp :\mathbb {R} \to \mathbb {R} }$ ${\displaystyle (\sin +\exp )(x)=\sin(x)+\exp(x)}$

Let be a non-empty set and a vector space over a field , then denotes (also or ) the set of all functions from to . The set becomes a vector space for and for scalars by the following two combinations: ${\displaystyle D}$ ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle V^{D}}$ ${\displaystyle \mathrm {Abb} (D,V)}$ ${\displaystyle F(D,V)}$ ${\displaystyle D}$ ${\displaystyle V}$ ${\displaystyle V^{D}}$ ${\displaystyle f,g\in V^{D}}$ ${\displaystyle \lambda \in K}$

• Addition: ${\displaystyle (f+g)\colon D\rightarrow V,x\mapsto f(x)+g(x)}$
• Scalar multiplication ${\displaystyle \lambda f\colon D\rightarrow V,x\mapsto \lambda \cdot f(x)}$

This vector space and the sub-vector spaces of are referred to as linear function space in the field of linear algebra . ${\displaystyle V^{D}}$ ${\displaystyle V^{D}}$

In the topology

In the topology is meant by a functional area a topological space whose elements are functions of an amount or a topological space in a topological space and whose topology on the topology of and and any additional structures, such as a metric or a uniform structure derived, is. The compact open topology is often used. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle Y}$

In 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 mappings from to . A linear function space in the field of functional analysis is a subspace of which is provided with a topological structure derived from. ${\displaystyle D}$ ${\displaystyle V}$ ${\displaystyle V^{D}}$ ${\displaystyle D}$ ${\displaystyle V}$ ${\displaystyle V^{D}}$ ${\displaystyle V}$

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. During this time the function spaces emerged of times continuously-differentiable functions, as well as the classical Lebesgue spaces of -integrable functions. In addition, the rooms with the continuous functions and the classic Hardy rooms are included in this phase. ${\displaystyle C^{k}}$ ${\displaystyle k}$ ${\displaystyle p}$

The second, the constructive phase, began with the publications of Sergei Lvowitsch Sobolew from the years 1935 to 1938, in which he introduced the (integer) Sobolew rooms named after him today . The theory of distributions emerged and new techniques, such as embedding theorems , were developed for solving partial differential equations. In this phase, function rooms were equipped with standards or quasi-standards . Important newly developed rooms of this time are the Zygmund rooms (or classes), the Slobodeckij rooms , the classic Besov rooms and the Bessel potential rooms . In the 1960s, the BMO room by Fritz John and Louis Nirenberg and the real Hardy rooms by Elias Stein and Guido Weiss were introduced.

The third phase, known as the systematic phase, began in the 1960s and clearly overlapped with the constructive phase. Here, the techniques of Fourier analysis were further developed and so-called maximum inequalities were examined. With the help of these tools, the Besov-Lebesgue spaces and the Lizorkin-Triebel spaces were developed. These two rooms can be embedded in the room of the temperature-controlled distributions . As their definitions suggest, these spaces are very closely intertwined with Fourier analysis. The modulation spaces follow a similar concept, but with congruent instead of dyadic overlaps . ${\displaystyle B_{p,q}^{s}}$ ${\displaystyle F_{p,q}^{s}}$ ${\displaystyle S'}$

Examples

topology

• If and are topological spaces , then one writes for the set of continuous functions . ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle {\mathcal {C}}(M,N)}$ ${\displaystyle f\colon M\to N}$
• If a metric is given, then it makes sense to speak of the set of restricted functions (even without topology ). Among other things, the notation is used for this figure . If a topology is also defined, one writes for the set of bounded continuous functions. On these rooms is through ${\displaystyle N}$ ${\displaystyle d}$ ${\displaystyle M}$ ${\displaystyle B(M,N)}$ ${\displaystyle M}$ ${\displaystyle {\mathcal {C}}_{b}(M,N)}$

${\displaystyle d_{\infty }\colon (f,g)\mapsto \sup _{x\in M}d(f(x),g(x))}$

defines a metric. Alternatively, there is also the metric

${\displaystyle d'_{\infty }\colon (f,g)\mapsto \min\{1,\sup _{x\in M}d(f(x),g(x))\}}$

possible. However, these two metrics produce the same open sets , so they can be treated equivalent.

• If the topologies are given on and by a pseudometric or a metric , then one writes for the set of uniformly continuous functions . If and are uniform spaces , then this notation describes the set of uniform-continuous functions, that is, those functions that respect the uniform structures. ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle {\mathcal {C}}_{u}(M,N)}$ ${\displaystyle M}$ ${\displaystyle N}$
• Is the body of real numbers or complex numbers , and is clear from the context in which the body reflect the functions, it is in the notation usually omitted, and it then writes short , or . ${\displaystyle N}$ ${\displaystyle {\mathcal {C}}(M)}$ ${\displaystyle {\mathcal {C}}_{b}(M)}$ ${\displaystyle {\mathcal {C}}_{u}(M)}$

Functional analysis

Most function spaces are examined in functional analysis. The following list is a selection of the rooms examined there. Let be the definition set of the examined functions. Then ${\displaystyle D}$

• ${\displaystyle {\mathcal {C}}^{p}(D)}$the space of -fold continuously differentiable functions with . If is compact , the space is relative to the usual norm ${\displaystyle p}$ ${\displaystyle p\in \mathbb {N} \cup \{0,\infty \}}$ ${\displaystyle D}$

${\displaystyle \|f\|_{{\mathcal {C}}^{p}(D)}=\sup _{k\leq p}\,\sup _{x\in D}|f^{(k)}(x)|}$

a Banach room . See differentiation class .

• ${\displaystyle {\mathcal {C}}^{p,\alpha }(D)}$the space of the -fold continuously differentiable functions, which are Hölder continuous with exponents . If it is compact, it is provided with the standard ${\displaystyle p}$ ${\displaystyle \alpha \in (0,1]}$ ${\displaystyle D}$ ${\displaystyle {\mathcal {C}}^{p,\alpha }(D)}$

${\displaystyle \|f\|_{C^{p,\alpha }}:=\sum _{|\beta |\leq p}{\sup _{x\in D}{\|(D^{\beta }f)(x)\|}}+\sum _{|\beta |=p}\sup _{x\neq y}{\frac {|(D^{\beta }f)(x)-(D^{\beta }f)(y)|}{|x-y|^{\alpha }}}}$

a Banach space, where is a multi-index . is also referred to as the space of Lipschitz's continuous functions. ${\displaystyle \beta }$ ${\displaystyle {\mathcal {C}}^{p,1}(D)}$

• ${\displaystyle C_{0}^{\infty }}$, or the space of the test functions . It contains all smooth functions with a compact carrier and is provided with the topology which is induced by the concept of convergence. A sequence converges in against if there is a compact term with for all j, and ${\displaystyle C_{c}^{\infty }}$ ${\displaystyle {\mathcal {D}}(D)}$ ${\displaystyle (\phi _{j})_{j\in J}}$ ${\displaystyle {\mathcal {D}}(D)}$ ${\displaystyle \phi }$ ${\displaystyle K\subset D}$ ${\displaystyle \operatorname {supp} (\phi _{j})\subset K}$

${\displaystyle \lim _{j\to \infty }\sup _{x\in K}\left|\partial _{x}^{\alpha }(\phi _{j}(x)-\phi (x))\right|=0}$

applies to all multi-indices . ${\displaystyle \alpha \in \mathbb {N} ^{n}}$

• ${\displaystyle L^{p}(D)}$the space of the -fold Lebesgue integrable functions (see L ^p ). This space does not consist of individual functions, but of equivalence classes of functions that differ only on a Lebesgue null set . For this reason, the standard ${\displaystyle p}$ ${\displaystyle p\geq 1}$ ${\displaystyle L^{p}}$

${\displaystyle \|f\|_{L^{p}(D)}=\left(\int _{D}|f(x)|^{p}\mathrm {d} x\right)^{1/p}}$

positive definite and therefore really a norm. Regarding this norm, the space on compact sets is also a Banach space. The special case L ² is even a Hilbert space . This space is often used in quantum mechanics. It is the space of the wave functions. The spaces for can be defined analogously, but these are not standardized spaces. ${\displaystyle L^{p}}$ ${\displaystyle 0<p<1}$ ${\displaystyle L^{p}}$

• ${\displaystyle L_{\mathrm {loc} }^{1}(D)}$the space of locally integrable functions . Be a measurable function. Locally integrable means that for all compact subsets the integral ${\displaystyle f\colon D\to \mathbb {R} }$ ${\displaystyle K\subset D}$

${\displaystyle \int _{K}|f(x)|\,\mathrm {d} x}$

is finite. Just like the spaces, the space consists of equivalence classes of functions. In particular, continuous functions and functions can be integrated locally. The space is needed when considering regular distributions . ${\displaystyle L^{p}}$ ${\displaystyle L_{\mathrm {loc} }^{1}(D)}$ ${\displaystyle L^{p}}$ ${\displaystyle L_{\mathrm {loc} }^{1}(\mathbb {R} )}$

• ${\displaystyle W^{k,p}(D)}$the space of weakly differentiable functions. It is called the Sobolev room . This space is often used as an approach space for solving differential equations. Because every continuously differentiable function is also weakly differentiable.

• ${\displaystyle S(\mathbb {R} ^{n})}$the space of functions that fall faster than any polynomial function . The set is called the Schwartz space , named after the French mathematician Laurent Schwartz of the same name . The space was constructed so that the Fourier transform is an isomorphism on it. The dual space of the Schwartz space is the space of the tempered distributions .

• By real or complex number sequences as pictures of after or conceives, can be understood as a function room and every vector space of sequences. ${\displaystyle \mathbb {N} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$

• ${\displaystyle {\mathcal {O}}(D)}$is the space of holomorphic functions. These functions can be differentiated any number of times, and their Taylor series converges to the output function. Holomorphic functions are often called analytical. Sometimes this space is also noted down . ${\displaystyle C^{\omega }(D)}$

• ${\displaystyle H^{p}(D)}$is the space of holomorphic, integrable functions, it is called Hardy space and is an analogue to space. The unit sphere is usually used as the definition set. ${\displaystyle L^{p}}$

Function rooms in theoretical computer science

Function spaces in connection with models of the lambda calculus are used here in particular . Its objects appear equally as functions, but also as their arguments and results. It is therefore desirable to have a subject area whose function space is isomorphic to itself, but this is not possible for cardinality reasons. Dana Scott was able to solve this problem in 1969 by restricting to continuous functions with regard to a suitable topology . Denotes the continuous functions of a complete partial order , then is . This form of function spaces is the subject of domain theory today . Later, a suitable function space could be found as a retraction of an object in a Cartesian closed category . ${\displaystyle D}$ ${\displaystyle D^{D}}$ ${\displaystyle D}$ ${\displaystyle D^{D}}$ ${\displaystyle D}$ ${\displaystyle [D\rightarrow D]}$ ${\displaystyle D\cong [D\rightarrow D]}$ ${\displaystyle D^{D}}$ ${\displaystyle D}$

literature

• Hans Triebel: Theory of function spaces. Birkhäuser Verlag, 1983, ISBN 3-7643-1381-1 .

Individual evidence

1. ^ J. Naas, HL Schmid: Mathematical dictionary. BG Teubner, Stuttgart 1979, ISBN 3-519-02400-4 .
2. ^ H. Heuser: Textbook of Analysis Part 1. 5th Edition. Teubner-Verlag, 1988, ISBN 3-519-42221-2 .
3. ^ Boto von Querenburg: Set theoretical topology . Springer-Verlag, Berlin / Heidelberg / New York 1976, ISBN 3-540-06417-6 , pp. 160 .
4. ^ ^{A b c} Hans Triebel: Theory of function spaces. Birkhäuser Verlag, 1983, ISBN 3-7643-1381-1 , pp. 33-35.
5. ^ Otto Forster , Thomas Szymczak: Exercise book for Analysis 2 . Tasks and solutions. 7th edition. 2011, ISBN 978-3-8348-1253-7 , pp. 5 and 39 f . (Proof only for ). ${\displaystyle p=0}$
6. ^ HP Barendregt: The Lambda Calculus. Elsevier, 1984, ISBN 0-444-87508-5 , p. 86.