HOME

TheInfoList



OR:

:''"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 bou ...
. 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 cons ...
in isolation is a boundaryless bounded set, while the half plane 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 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 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 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 mathe ...
. 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 in ...
is a closed but is not bounded (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 ...
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

A
subset In mathematics, set ''A'' is a subset of a set ''B'' if all 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 unequal, then ''A'' is a proper subset of ...
''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 setti ...
(''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 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 Britis ...
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'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
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 space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
s, 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 is ...
ness. If the topology of the topological vector space is induced by a
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 mathem ...
which is homogeneous, as in the case of a metric induced by the norm of
normed vector spaces In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
, 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 bina ...
. 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 In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an elem ...
. 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 logi ...
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 In mathematics, given two preordered sets A and B, the product order (also called the coordinatewise orderDavey & Priestley, ''Introduction to Lattices and Order'' (Second Edition), 2002, p. 18 or componentwise order) is a partial ordering o ...
. 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 a subclass of the class of all ordinal numbers.


See also

* Bounded function * Local boundedness *
Order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article int ...
*
Totally bounded 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� ...


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