Limited amount

A bounded set with upper and lower bounds.

A limited amount with Supremum.

Bounded sets are considered in various areas of mathematics . The set is then referred to as the restricted (up or down) set. First of all, this means that all elements of the set are not below or above a certain limit with regard to an order relation . More precisely, one then speaks of the fact that the set is restricted with regard to the relation (upwards or downwards). The terms upper and lower bound are described in detail in the article Supremum . ${\displaystyle \leq }$ ${\displaystyle \leq }$

The term is used much more often in a figurative sense. Then a set is called bounded (upwards) if a distance function between its elements, which usually has the non-negative real numbers as a set of values, only accepts values ​​not above a certain real number. Here the downward restriction (namely by 0) is mostly self-evident, so we simply speak of a restricted set here. More precisely, one would have to say: The set is limited in terms of the distance function (and the natural arrangement of its value stock). ${\displaystyle d}$ ${\displaystyle d}$

There is also the concept of a function that is restricted (upwards or downwards) . This is to be understood as a function whose image set (as a subset of a semi-ordered set) has the corresponding property, or in the figurative sense: the set of images of the function has the corresponding restriction property with regard to a distance function.

Definitions

Restriction with regard to an order relation

Let be a set semi-ordered by the relation and a subset of . ${\displaystyle M}$ ${\displaystyle \leq }$ ${\displaystyle S}$ ${\displaystyle M}$

• One element is an upper bound of if: . That means: All elements of are less than or equal to the upper bound . If such an upper bound exists, it is called upwardly bounded (with regard to the relation ). ${\displaystyle b\in M}$ ${\displaystyle S}$ ${\displaystyle \forall x\in S\colon x\leq b}$ ${\displaystyle S}$ ${\displaystyle b}$ ${\displaystyle b}$ ${\displaystyle S}$ ${\displaystyle \leq }$
• An element is a lower bound of if: . That means: All elements of are greater than or equal to the lower bound . If such a lower bound exists, it is called downwardly bounded (with respect to the relation ). ${\displaystyle a\in M}$ ${\displaystyle S}$ ${\displaystyle \forall x\in S\colon a\leq x}$ ${\displaystyle S}$ ${\displaystyle a}$ ${\displaystyle a}$ ${\displaystyle S}$ ${\displaystyle \leq }$
• A set that is bounded both upwards and downwards in this sense is called a bounded set (with respect to the relation ). ${\displaystyle S}$ ${\displaystyle \leq }$
• A set that is not limited is called unlimited.
• A function in a semi-ordered set is called bounded upwards or downwards if there is an upper or lower bound for the image set . Is restricted both upwards and downwards, it is called restricted, otherwise unrestricted. ${\displaystyle f\colon X\to M}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle S=f(X)=\{f(x)\mid x\in X\}}$ ${\displaystyle f}$ ${\displaystyle f}$

Transfer to sets on which a distance function is defined

The terms restricted and unrestricted , which are thus defined for a semi-ordered set, are now also used in a figurative sense for sets with a distance function, if the values ​​that this function assumes have the corresponding bounds in the ordered image set (mostly non-negative real numbers) has (or does not have).

Transfer to functions for which a range of values ​​is defined

Let be a set and a distance function on , an arbitrary set. A function is called bounded (with respect to the distance function ) if the set in is bounded, otherwise unbounded. ${\displaystyle X}$ ${\displaystyle d\colon X\times X\to \mathbb {R} }$ ${\displaystyle X}$ ${\displaystyle N}$ ${\displaystyle f\colon N\to X}$ ${\displaystyle d}$ ${\displaystyle \left\{d(f(n_{1}),f(n_{2}))\mid n_{1},n_{2}\in N\right\}}$ ${\displaystyle \mathbb {R} }$

Analysis

The number of images of the function shown is limited, so the function is also limited.

In analysis , a subset of real numbers is called upwardly bound if and only if there is a real number with for all out . Every such number is called the upper bound of . The terms below restricted and lower bound are defined analogously. ${\displaystyle S}$ ${\displaystyle k}$ ${\displaystyle k\geq s}$ ${\displaystyle s}$ ${\displaystyle S}$ ${\displaystyle k}$ ${\displaystyle S}$

The set is called restricted if it is bounded above and below it. Hence a set is bounded if it lies in a finite interval. ${\displaystyle S}$

This results in the following relationship: A subset of the real numbers is restricted if and only if there is a real number such that from applies to all . It is then said that the open sphere (i.e. an open interval) is around 0 with radius . ${\displaystyle S}$ ${\displaystyle R}$ ${\displaystyle |x|<R}$ ${\displaystyle x}$ ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle R}$

If they exist, the smallest upper bound is called the supremum of , the largest lower bound is called the infimum. ${\displaystyle S}$

A function is called restricted to if its set of images is a restricted subset of . ${\displaystyle f\colon X\to \mathbb {R} }$ ${\displaystyle X}$ ${\displaystyle f(X)}$ ${\displaystyle \mathbb {R} }$

A subset of the complex numbers is called restricted if the amounts of each element of do not exceed a certain limit. That is, the quantity is contained in the closed circular disk . A complex-valued function is called bounded if its set of images is bounded. ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle R}$ ${\displaystyle S}$ ${\displaystyle K_{R}(0)=\{z\in \mathbb {C} :|z|\leq R\}}$

Correspondingly, the term is defined in the -dimensional vector spaces or : A subset of these spaces is called restricted if the norm of its elements does not exceed a common limit. This definition is independent of the special norm, since all norms in finite-dimensional normalized spaces lead to the same notion of limitation. ${\displaystyle n}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {C} ^{n}}$

Metric spaces

Limited amount (above) and unlimited amount (below)

A set from a metric space is called bounded if it is contained in a closed sphere of finite radius; H. when and exist, such that for all of the following applies: . ${\displaystyle S}$ ${\displaystyle (M,d)}$ ${\displaystyle x\in M}$ ${\displaystyle r>0}$ ${\displaystyle s}$ ${\displaystyle S}$ ${\displaystyle d(x,s)\leq r}$

Functional analysis

Bounded sets in topological vector spaces

A subset of a topological vector space is called bounded if there is a for every neighborhood of 0 such that . ${\displaystyle S}$ ${\displaystyle U}$ ${\displaystyle k>0}$ ${\displaystyle S\subseteq kU}$

If a locally convex space is, its topology is given by a set of semi-norms . The boundedness can then be characterized by semi-norms as follows: is restricted if and only if for all semi-norms . ${\displaystyle E}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle S\subset E}$ ${\displaystyle \textstyle \sup _{x\in S}p(x)<\infty }$ ${\displaystyle p\in {\mathcal {P}}}$

Examples of restricted quantities

• Compact quantities are limited.
• The unit sphere in an infinitely dimensional normalized space is restricted but not compact.
• Let be the vector space of all finite sequences, i.e. H. of all consequences , so that for almost everyone . Keep going . Then respect. By limited defined standard, but not respect. By defined standard. ${\displaystyle c_{00}}$ ${\displaystyle (x_{n})_{n}}$ ${\displaystyle x_{n}\in \mathbb {R} ,x_{n}=0}$ ${\displaystyle n}$ ${\displaystyle S:=\{(x_{n})_{n}\in c_{00}:|x_{n}|\leq 1\,\forall n\}}$ ${\displaystyle S}$ ${\displaystyle \textstyle \|(x_{n})_{n}\|_{\infty }=\sup _{n\in \mathbb {N} }|x_{n}|}$ ${\displaystyle \|(x_{n})_{n}\|_{1}=\sum _{n\in \mathbb {N} }|x_{n}|}$
• If one considers the locally convex topology defined by the semi-norms in the space of finite sequences of the previous example , then it is restricted. This amount is not limited for either of the two standards mentioned. ${\displaystyle c_{00}}$ ${\displaystyle p_{m}((x_{n})_{n}):=|x_{m}|}$ ${\displaystyle S:=\{(x_{n})_{n}\in c_{00}:|x_{n}|\leq n\,\forall n\}}$

Permanent properties

• Subsets of restricted sets are restricted.
• Finite unions of bounded sets are bounded.
• The topological closure of a bounded set is bounded.
• Are and limited, so too . ${\displaystyle S}$ ${\displaystyle T}$ ${\displaystyle S+T:=\{s+t\mid s\in S,t\in T\}}$
• A continuous, linear mapping between locally convex spaces maps limited sets to limited sets (see also: Bornological space ).
• If locally convex, then the convex hull and the absolutely convex hull of a bounded set are again bounded. ${\displaystyle E}$

literature

• Bernd Aulbach: Analysis. Volume 1. University, Augsburg 2001.
• Harro Heuser : Textbook of Analysis. Part 1. 5th revised edition. Vieweg + Teubner, Wiesbaden 1988, ISBN 3-519-42221-2 .