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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. Conversely a set that is not bounded is called unbounded. Bounded sets are a natural way to define a locally convex polar topologies on the vector spaces in a dual pair, as the 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. == Definition == Given a topological vector space (''X'',τ) over a field ''F'', ''S'' is called bounded if for every neighborhood ''N'' of the zero vector there exists a scalar α such that : with :. This is equivalent〔Schaefer 1970, p. 25.〕 to the condition that ''S'' is absorbed by every neighborhood of the zero vector, i.e., that for all neighborhoods ''N'', there exists ''t'' such that :. In locally convex topological vector spaces the topology τ of the space can be specified by a family ''P'' of semi-norms. 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 spaces (''X'',''p'') with ''p'' a semi norm of ''P''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bounded set (topological vector space)」の詳細全文を読む スポンサード リンク
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