:''"Bounded" and "boundary" are distinct concepts; for the latter see

boundary (topology) In topology and mathematics in general, the boundary of a subset of a topological space is the set of points in the closure of not belonging to the interior of . An element of the boundary of is called a boundary point of . The term bound ...

. A

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...

in isolation is a boundaryless bounded set, while the

half plane In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional Euclidean space. If the space is two-dimensional, then a half-space is called a half-plane (open or closed). A half-space in a one-dimensional s ...

is unbounded yet has a boundary. In

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...

and related areas of mathematics, a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

is called bounded if it is, in a certain sense, of finite measure. Conversely, a set which is not bounded is called unbounded. The word 'bounded' makes no sense in a general topological space without a corresponding metric. A bounded set is not necessarily a

closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...

and vise versa. For example, a subset ''S'' of a 2-dimensional real space R^''2'' constrained by two parabolic curves ''x''² + 1 and ''x''² - 1 defined in a

Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured i ...

is a closed but is not bounded (unbounded).

Definition in the real numbers

A set ''S'' of real numbers is called ''bounded from above'' if there exists some real number ''k'' (not necessarily in ''S'') 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.

Definition in a metric space

A subset ''S'' of a

metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...

(''M'', ''d'') is bounded if there exists ''r'' > 0 such that for all ''s'' and ''t'' in ''S'', we have d(''s'', ''t'') < ''r''. The metric space (''M'', ''d'') is a ''bounded'' metric space (or ''d'' is a ''bounded'' metric) if ''M'' is bounded as a subset of itself. *

Total boundedness In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size� ...

implies boundedness. For subsets of R^''n'' the two are equivalent. *A metric space is compact if and only if it is

complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...

and totally bounded. *A subset of Euclidean space R^''n'' is compact if and only if it is

closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

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 homogeneous, as in the case of a metric induced by the

norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...

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 In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...

. 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 interval. Note that this is not just a property of the set ''S'' but also 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 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 In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore o ...

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 number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least ...

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 a subclass of the class of all ordinal numbers.

References

* *{{cite book , first=Robert D. , last=Richtmyer , author-link=Robert D. Richtmyer , title=Principles of Advanced Mathematical Physics , publisher=Springer , location=New York , year=1978 , isbn=0-387-08873-3 Mathematical analysis Functional analysis Order theory