Bounded set

The property of boundedness is assigned in different areas of mathematics a lot. The amount is then referred to as limited ( up or down ) amount. This is initially meant to be that all elements of the set with respect to an order relation is not below or not above a certain limit. More precisely, one speaks then assume that the set is bounded with respect to the relation ( up or down ). The terms upper and lower bounds are described in detail in the article supremum.

Much more frequently, the term is used in a figurative sense. Then say a lot of limited (up) if a distance function between their elements, as the range of values ​​usually has the non-negative real numbers, only values ​​does not accept above a certain real number. Here the boundedness understands down ( namely 0) mostly by itself, so people will just spoken only by a limited amount. More should be said: the amount with respect to the distance function (and the natural arrangement of whose range of values ​​) is limited.

In addition, there is the concept of a limited ( up or down ) function. Below is a function to understand their image set ( as a subset of a partially ordered set ) has the corresponding property or figuratively: The number of images the function has with respect to a distance function, the corresponding boundedness property.

  • 4.1 Limited quantity in topological vector spaces
  • 4.2 Examples of limited quantities
  • 4.3 Permanenzeigenschaften
  • 4.4 Restricted pictures
  • 5.1 Uniform boundedness
  • 5.2 Pointwise boundedness
  • 5.3 Examples

Definitions

Boundedness with respect to an order relation

Let be a partially ordered by the relation set and a subset of the.

  • An element is called an upper bound of if:. This means that all elements of less than or equal to the upper bound. If such an upper bound exists, is called bounded above ( with respect to the relation).
  • An element is called lower bound if and only if. This means that all elements of greater than or equal to the lower bound. If such a lower bound exists, is called bounded below ( with respect to the relation).
  • A lot of that is limited in this sense, both up and down is called ( with respect to the relation) as a bounded set.
  • A lot of that is not restricted means unlimited.
  • A function into a partially ordered set is called up or limited down when there is an upper or lower bound for the image set. If both up and down limited, called limited, or unlimited.

Transfer to quantities at which a distance function is defined

Limited the terms and unlimited, which are defined on a partially ordered set, are now used in a figurative sense to quantities with a distance function if the values ​​taken by this function in the parent image volume (normally non-negative real numbers ), the corresponding barriers has (or has not ).

Transfer to functions on the set of values ​​a distance function is defined

Be a quantity and a distance function on an arbitrary set. A function is called bounded (with respect to the distance function), if the amount is limited, or unlimited.

Analysis

In calculus, ie a subset of the real numbers bounded above if and only if there is a real number such that for all out. Each such number is called an upper bound of. The terms bounded below and lower bound are defined analogously.

The quantity is called bounded if it is bounded above and bounded below. Consequently, a set is bounded if it lies in a finite interval.

This results in the relationship: A subset of the real numbers is bounded if and only if there exists a real number, such that for all. They say then, lay in the open ball (that is, an open interval ) around 0 with radius.

In the case of their existence is called the least upper bound of the supremum, the greatest lower bound the infimum.

A function is called on restricted if its image set is a bounded subset of.

A subset of the complex numbers is called bounded if the amounts of each element of a certain barrier does not exceed. That is, the amount is in the closed disc. A complex-valued function is called bounded if its image set is limited.

The term is defined in the corresponding full -dimensional vector spaces and a subset of these spaces is called bounded if the norm of its elements does not exceed a common barrier. This definition is independent of the particular standard, since all norms in finite-dimensional normed spaces lead to the same limitations of term.

Metric spaces

A set S of a metric space (M, d) is called bounded if it is contained in a ball of finite radius, ie when an x in M and r > 0 exist such that s of S is valid for all.

Functional Analysis

Limited quantity in topological vector spaces

A subset of a topological vector space is called bounded if there is any neighborhood of 0, so that is true.

If E is a locally convex space, whose topology is given by a set of semi-norms. The limitations can then be characterized as follows by semi-norms: is bounded if and only if for all semi-norms.

Examples of limited quantities

  • Compact quantities are limited.
  • The unit sphere in an infinite-dimensional normed space is limited but not compact.
  • Be the vector space of all finite sequences, that is, of all sequences such that for almost all n Let further. Then S is limited by regarding the defined standard, but not with respect to the norm defined by.
  • Looking on the space of finite sequences of the previous example defined by the semi-norms locally convex topology, so is limited. This amount is limited for any of the two standards mentioned above.

Permanenzeigenschaften

  • Subsets of limited quantities are limited.
  • Finite unions of bounded sets are bounded.
  • The topological degree of a limited quantity is limited.
  • S and T are bounded, then also
  • A continuous, linear map between locally convex spaces is limited quantities on bounded sets (see also: bornological space ).
  • If E is locally convex, then the convex hull and the absolutely convex hull of a bounded set is again limited.

Restricted pictures

Are and topological vector spaces, it means a limited picture when the image is limited every bounded subset.

Are and normed spaces, so this condition is equivalent to saying that there exists a constant such that

Applies. The amount of these is bounded from below and no upper limit, so the infimum of this set exists - it is identical to the operator norm on. It can be shown that each bounded linear operator between the normalized space is continuous and vice versa.

For further use of the concept of boundedness in the context of semigroups on Banach spaces see strongly continuous semigroup.

Limited functions and uniform boundedness

Uniform boundedness

The term uniform boundedness is applied only to sets of functions, ie, sets of functions with the same definition and the same amount of values. Usually one speaks of families of functions or, if the family is countable infinity of a sequence of functions.

Be an arbitrary set. Then a family of on -defined, real-valued functions is called uniformly bounded if there is a real number for which:. That is, from a common upper bound on the values ​​of the amounts of all functions.

Obviously, a family of functions can be at most then uniformly bounded if every single feature of the family is limited. For each function, therefore there is the supremum norm. A family of functions is now uniformly bounded if and only if it is limited as a set of functions with respect to the supremum norm.

This is generalized to vector valued functions: where any quantity, a real or complex normed space with the standard. It denotes the set of functions defined on, which are limited in respect to the norm, as and leads to a with a standard, which in turn makes for a normed space. Then a family of functions defined on is bounded if and evenly if it is a subset of, and as a subset of is restricted.

Pointwise boundedness

A family of functions on a set whose range of values ​​is a normed space, ie, pointwise bounded if for each point the amount is limited. Thus, the amount of all the values ​​to be adopted in the place of any function of the family.

Note:

  • A uniformly bounded family is necessary pointwise limited.
  • A pointwise bounded family can contain unlimited functions.
  • Every uniformly bounded family consists of bounded functions, but not every family of bounded functions is uniformly bounded.
  • A family of bounded functions need not be pointwise limited.

Examples

All families in the examples are defined as sequences of functions. The family is always here.

Mathematical analysis Closed set Operatornorm Harro Heuser
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