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| {{for|bounded sets in general|bounded set}}
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| In [[functional analysis]] and related areas of [[mathematics]], a [[Set (mathematics)|set]] in a [[topological vector space]] is called '''bounded''' or '''von Neumann bounded''', if every [[neighbourhood (mathematics)|neighborhood]] of the [[zero vector]] can be ''inflated'' to include the set. Conversely a set which is not bounded is called '''unbounded'''.
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| Bounded sets are a natural way to define a [[locally convex topology|locally convex]] [[polar topology|polar topologies]] on the [[vector space]]s in a [[dual pair]], as the [[polar set|polar]] of a bounded set is an [[absolutely convex]] and [[absorbing set]]. The concept was first introduced by [[John von Neumann]] and [[Andrey Kolmogorov]] in 1935.
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| == Definition ==
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| Given a [[topological vector space]] (''X'',τ) over a [[field (mathematics)|field]] ''F'', ''S'' is called '''bounded''' if for every neighborhood ''N'' of the zero vector there exists a [[scalar (mathematics)|scalar]] α such that
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| :<math>S \subseteq \alpha N</math> | |
| with
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| :<math>\alpha N := \{ \alpha x \mid x \in N\}</math>.
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| In other words a set is called bounded if it is [[absorbing set|absorbed]] by every [[neighbourhood (mathematics)|neighborhood]] of the zero vector.
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| In [[locally convex topological vector space]]s the topology τ of the space can be specified by a family ''P'' of [[semi-norm]]s. An equivalent characterization of bounded sets in this case is, a set ''S'' in (''X'',''P'') is bounded if and only if it is bounded for all [[semi normed space]]s (''X'',''p'') with ''p'' a semi norm of ''P''.
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| == Examples and nonexamples ==
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| * Every finite set of points is bounded
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| * The set of points of a [[Cauchy sequence]] is bounded, the set of points of a Cauchy [[net (mathematics)|net]] need not to be bounded.
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| * Every [[relatively compact set]] in a topological vector space is bounded. If the space is equipped with the [[weak topology (polar topology)|weak topology]] the converse is also true.
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| * A (non null) subspace of a Hausdorff topological vector space is '''not''' bounded
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| == Properties ==
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| * The [[closure (topology)|closure]] of a bounded set is bounded.
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| * In a locally convex space, the [[convex envelope]] of a bounded set is bounded. (Without local convexity this is false, as the <math>L^p</math> spaces for <math>0<p<1</math> have no nontrivial open convex subsets.)
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| * The finite [[union (set theory)|union]] or finite sum of bounded sets is bounded.
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| * [[Continuous linear mapping]]s between topological vector spaces preserve boundedness.
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| * A [[locally convex space]] is [[seminormable]] [[if and only if]] there exists a bounded neighbourhood of zero.
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| * The polar of a bounded set is an absolutely convex and absorbing set.
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| * A set ''A'' is bounded if and only if every [[countable]] subset of ''A'' is bounded
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| == Generalization ==
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| The definition of bounded sets can be generalized to [[topological module]]s. A subset ''A'' of a topological module ''M'' over a [[topological ring]] ''R'' is bounded if for any neighborhood ''N'' of ''0<sub>M</sub>'' there exists a neighborhood ''w'' of 0<sub>''R''</sub> such that ''w A ⊂ N''.
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| == See also ==
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| *[[Totally bounded space]]
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| *[[Local boundedness]]
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| *[[bounded function]]
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| *[[bounding point]]
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| == References ==
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| * {{cite book |last=Robertson |first=A.P. |coauthors= W.J. Robertson |title= Topological vector spaces |series=Cambridge Tracts in Mathematics |volume=53 |year=1964 |publisher= [[Cambridge University Press]] | pages=44–46 }}
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| * {{cite book | author=H.H. Schaefer | title=Topological Vector Spaces | publisher=[[Springer-Verlag]] | series=[[Graduate Texts in Mathematics|GTM]] | volume=3 | date=1970 | isbn=0-387-05380-8 | pages=25–26 }}
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| {{Functional Analysis}}
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| [[Category:Topological vector spaces]]
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| [[nl:Begrensdheid]]
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