Bounded set

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Bounded set

"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology)
Boundary (topology)

In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S....
. A circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.

In mathematical analysis
Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of mathematics most explicitly concerned with the notion of a limit , whether the limit of a sequence or the limit of a function....
and related areas of mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, 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
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s is called bounded from above if there is a real number k such that k = s for all s in S.

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Encyclopedia

"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology)
Boundary (topology)

In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S....
. A circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.
In mathematical analysis
Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of mathematics most explicitly concerned with the notion of a limit , whether the limit of a sequence or the limit of a function....
and related areas of mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, 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
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
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
Interval (mathematics)

In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set....
.

Metric space

A subset
Subset

In mathematics, especially in set theory, a Set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide....
S of a metric space
Metric space

In mathematics, a metric space is a Set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space....
(M, d) is bounded if it is contained in a ball
Ball (mathematics)

In mathematics, a ball is the inside of a sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions, and for metric spaces in general....
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 Rn the two are equivalent.
A metric space is compact
Compact space

In mathematics, a topological space is called compact if each of its open covers has a finite set subcover.Note: Some authors such as Nicolas Bourbaki use the term "quasi-compact" for this instead, and reserve the term "compact" for topological spaces that are both Hausdorff spaces and "quasi-compact"....
if and only if it is complete and totally bounded.
A subset of Euclidean space
Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
Rn is compact if and only if it is closed
Closed set

In topology and related branches of mathematics, a closed set is a Set whose complement is open set....
and bounded.

Boundedness in topological vector spaces

In topological vector space
Topological vector space

In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. As the name suggests the space blends a topology with the algebraic concept of a vector space....
s, 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
Metric (mathematics)

In mathematics, a metric or distance function is a function which defines a distance between elements of a Set . A set with a metric is called a metric space....
which is homogenous, as in the case of a metric induced by the norm
Norm (mathematics)

In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector....
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
Partially ordered set

In mathematics, especially order theory, a partially ordered set formalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a 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 . 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
Greatest element

In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S....
. 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 of the order on P is not necessarily a bounded poset.

A subset S of Rn is bounded with respect to the Euclidean distance
Euclidean distance

In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem....
if and only if it bounded as subset of Rn with the product order
Product order

In mathematics, given two ordered sets A and B, one can induce an ordering on the Cartesian product A × B. Giventwo pairs and in A × B, one sets...
. However, S may be bounded as subset of Rn with the lexicographical order
Lexicographical order

In mathematics, the lexicographic or lexicographical order, , is a natural order theory structure of the Cartesian product of two ordered sets....
, but not with respect to the Euclidean distance.

A class of ordinal number
Ordinal number

In set theory, an ordinal number, or just ordinal, is the order type of a well-order. They are usually identified with hereditarily transitive sets....
s is said to be unbounded, or cofinal
Cofinal (mathematics)

In mathematics, let A be a set and let ≤ be a binary relation on A. Then a subset B of A is said to be cofinal if it satisfies the following condition:This definition is most commonly applied when B is a partially ordered set or directed set under the relation ≤....
, 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
Bounded function

In mathematics, a function f defined on some Set X with real number or complex number values is called bounded, if the set of its values is bounded set....
Local boundedness
Local boundedness

In mathematics, a function is locally bounded, if it is bounded function around every point. A Family of functions is locally bounded, if for any point in their domain all the functions are bounded around that point and by the same number....
Order theory
Order theory

Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" or "precedes" another....
Totally bounded