# Limited amount

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 .

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).

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 .

- 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 ). - 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 ). - A set that is bounded both upwards and downwards in this sense is called a bounded set (with respect to the relation ).
- 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.

### 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.

## Analysis

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.

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.

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 .

If they exist, the smallest upper bound is called the supremum of , the largest lower bound is called the infimum.

A function is called restricted to if its set of images is a restricted subset of .

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.

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.

## Metric spaces

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: .

## 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 .

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 .

### 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.
- 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.

### 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 .
- 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.

## 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 .