Bounded set

Bounded set

In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. Conversely a set which is not bounded is called unbounded.

Definition

A set "S" of real numbers is called "bounded from above" if there is a real number "k" such that "k" ≥ " s" for all "s" in "S". The number "k" is called an upper bound of "S". The terms "bounded from below" and lower bound are similarly defined.

A set "S" is bounded if it has both upper and lower bounds. Therefore, a set of real numbers is bounded if it is contained in a finite interval.

Metric space

A subset "S" of a metric space ("M", "d") is bounded if it is contained in a ball of finite radius, i.e. if there exists "x" in "M" and "r" > 0 such that for all "s" in "S", we have d("x", "s") < "r". "M" is a "bounded" metric space (or "d" is a "bounded" metric) if "M" is bounded as a subset of itself.

*Total boundedness implies boundedness. For subsets of R"n" the two are equivalent.
*A metric space is compact if and only if it is complete and totally bounded.
*A subset of Euclidean space R"n" is compact if and only if it is closed and bounded.

Boundedness in topological vector spaces

In topological vector spaces, a different definition for bounded sets exists which is sometimes called von Neumann boundedness. If the topology of the topological vector space is induced by a metric which is homogenous, as in the case of a metric induced by the norm of normed vector spaces, then the two definitions coincide.

Boundedness in order theory

A set of real numbers is bounded if and only if it has an upper and lower bound. This definition is extendable to subsets of any partially ordered set. Note that this more general concept of boundedness does not correspond to a notion of "size".

A subset "S" of a partially ordered set "P" is called bounded above if there is an element "k" in "P" such that "k" ≥ "s" for all "s" in "S". The element "k" is called an upper bound of "S". The concepts of bounded below and lower bound are defined similarly. (See also upper and lower bounds.)

A subset "S" of a partially ordered set "P" is called bounded if it has both an upper and a lower bound, or equivalently, if it is contained in an ml|Interval_%28mathematics%29|Intervals_in_order_theory|interval. Note that this is not just a property of the set "S" but one of the set "S" as subset of "P".

A bounded poset "P" (that is, by itself, not as subset) is one that has a least element and a greatest element. Note that this concept of boundedness has nothing to do with finite size, and that a subset "S" of a bounded poset "P" with as order the ml|Binary_relation|Restriction|restriction of the order on "P" is not necessarily a bounded poset.

A subset "S" of R"n" is bounded with respect to the Euclidean distance if and only if it bounded as subset of R"n" with the product order. However, "S" may be bounded as subset of R"n" with the lexicographical order, but not with respect to the Euclidean distance.

A class of ordinal numbers is said to be unbounded, or cofinal, when given any ordinal, there is always some element of the class greater than it. Thus in this case "unbounded" does not mean unbounded by itself but unbounded as subclass of the class of all ordinal numbers.

See also

*Bounded function
*Local boundedness
*Order theory
*Totally bounded


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • Bounded set (topological vector space) — In functional analysis and related areas of mathematics, a set in a topological vector space is called bounded or von Neumann bounded, if every neighborhood of the zero vector can be inflated to include the set. Conversely a set which is not… …   Wikipedia

  • Bounded (set theory) — In set theory a subset X of a limit ordinal lambda; is said to be bounded if its supremum is less than lambda;. If it is not bounded, it is unbounded, that is:sup(X)= lambda.,For functions, bounded or unboundedness refers to the range of the… …   Wikipedia

  • Bounded operator — In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded by the same number, over all non zero… …   Wikipedia

  • Bounded mean oscillation — In harmonic analysis, a function of bounded mean oscillation, also known as a BMO function, is a real valued function whose mean oscillation is bounded (finite). The space of functions of bounded mean oscillation (BMO), is a function space that,… …   Wikipedia

  • Bounded variation — In mathematical analysis, a function of bounded variation refers to a real valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a continuous function of a… …   Wikipedia

  • Bounded function — In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a number M >0 such that :|f(x)|le M for all x in X .Sometimes, if f(x)le A for all …   Wikipedia

  • Bounded quantifier — In the study of formal theories in mathematical logic, bounded quantifiers are often added to a language. These are two quantifiers in addition to forall and exists. They are motivated by the fact that determining whether a sentence with only… …   Wikipedia

  • Bounded complete poset — In the mathematical field of order theory, a partially ordered set is bounded complete if all of its subsets which have some upper bound also have a least upper bound. Such a partial order can also be called consistently complete, since any upper …   Wikipedia

  • Bounded deformation — In mathematics, a function of bounded deformation is a function whose distributional derivatives are not quite well behaved enough to qualify as functions of bounded variation, although the symmetric part of the derivative matrix does meet that… …   Wikipedia

  • bounded — adjective Of a set, that it is capable of being included within a ball of finite radius. A compact set must be bounded …   Wiktionary

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”