mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...

and related areas of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 all of its points are within a certain distance of each other. 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 Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathema ...

. '' Boundary'' is a distinct concept: for example, 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 const ...

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 ...

is unbounded yet has a boundary. 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 cl ...

and vice 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 in t ...

is closed by the curves but not bounded (so unbounded).

Definition in the real numbers

A set ''S'' of

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...

s 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

subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...

''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 settin ...

(''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 Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

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 Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

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. This is also called the Heine-Borel theorem.

Boundedness in topological vector spaces

In topological vector spaces, a different definition for bounded sets exists which is sometimes called

von Neumann bounded 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. A set that is not bounded i ...

ness. If the topology of the topological vector space is induced by a

which is

homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...

, 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 envir ...

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 (mathematics), set. A poset consists of a set toget ...

. 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 In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an element ...

.) 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 Restriction, restrict or restrictor may refer to: Science and technology * restrict, a keyword in the C programming language used in pointer declarations * Restriction enzyme, a type of enzyme that cleaves genetic material Mathematics and log ...

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, therefor ...

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 In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a ...

, 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 n ...

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