- Bounded set
In
mathematical analysis and related areas ofmathematics , 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 number s 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 ametric 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 compactif and only if it is complete and totally bounded.
*A subset ofEuclidean space R"n" is compact if and only if it is closed and bounded.Boundedness in topological vector spaces
In
topological vector space s, a different definition for bounded sets exists which is sometimes calledvon Neumann bounded ness. 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 ofnormed 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 theproduct order . However, "S" may be bounded as subset of R"n" with thelexicographical order , but not with respect to the Euclidean distance.A class of
ordinal number s 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.