Bounded set (nonfiction)

In mathematical analysis and related areas of mathematics, 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.

The word bounded makes no sense in a general topological space without a corresponding metric.

Definition

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.

Metric space

A subset S of a metric 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, 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 Rn the two are equivalent.
• A metric space is compact if and only if it is complete and totally bounded.
• A subset of Euclidean space Rn is compact if and only if it is closed 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 (see Bounded set (topological vector space). 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 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. 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 Rn is bounded with respect to the Euclidean distance if and only if it bounded as subset of Rn with the product order. However, S may be bounded as subset of Rn 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

• Ball (mathematics) (nonfiction)
• Bounded function (nonfiction)
• Bounded set (topological vector space) (nonfiction)
• Closed set (nonfiction)
• Compact space (nonfiction)
• Complete metric space (nonfiction)
• Euclidean distance (nonfiction)
• Euclidean space (nonfiction)
• Greatest and least elements (nonfiction)
• Interval (mathematics) (nonfiction)
• Local boundedness (nonfiction)
• Mathematical analysis (nonfiction)
• Metric (mathematics) (nonfiction)
• Metric space (nonfiction)
• Norm (mathematics) (nonfiction) - a function that assigns a strictly positive length or size to each vector in a vector space (except for the zero vector, which is assigned a length of zero).
• Normed vector space (nonfiction) - 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" in the real world.
• Order theory (nonfiction)
• Partially ordered set (nonfiction) - (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 relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order."
• Real number (nonfiction)
• Subset (nonfiction)
• Topological vector space (nonfiction) - a vector space (an algebraic structure) which is also a topological space, the latter thereby admitting a notion of continuity. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence. The elements of topological vector spaces are typically functions or linear operators acting on topological vector spaces, and the topology is often defined so as to capture a particular notion of convergence of sequences of functions. Hilbert spaces and Banach spaces are well-known examples.
• Totally bounded (nonfiction)
• Totally bounded space (nonfiction)
• Upper and lower bounds (nonfiction)

References

• Bartle, Robert G.; Sherbert, Donald R. (1982). Introduction to Real Analysis. New York: John Wiley & Sons. ISBN 0-471-05944-7.
• Richtmyer, Robert D. (1978). Principles of Advanced Mathematical Physics. New York: Springer. ISBN 0-387-08873-3.