Bounded set
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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.
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Definition
A set S of real numbers is called bounded 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 below and lower bound are similarly defined.
A set S is bounded if it is bounded both above and below. 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 is a bounded metric space (or d is a bounded metric) if M is bounded as a subset of itself. Properties which are similar to boundedness but stronger, that is they imply boundedness, are total boundedness and compactness.
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 homogenous, as in the case of a metric induced by the norm of normed vector spaces, then the two definitions coincide.
Boundedness in order theory
The concepts of upper bound and lower bound can be extended to ordered sets. A totally ordered set S is called bounded above if there is an element k 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
he:מרחב חסום hu:Korlátos halmaz ja:有界 pl:Zbiór ograniczony pt:Limitado