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In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field ''K'' with an absolute value function | |) is a set ''S'' such that for all scalars α with |α| ≤ 1 : where : The balanced hull or balanced envelope for a set ''S'' is the smallest balanced set containing ''S''. It can be constructed as the intersection of all balanced sets containing ''S''. == Examples == * The open and closed balls centered at 0 in a normed vector space are balanced sets. * Any subspace of a real or complex vector space is a balanced set. * The cartesian product of a family of balanced sets is balanced in the product space of the corresponding vector spaces (over the same field ''K''). * Consider ℂ, the field of complex numbers, as a 1-dimensional vector space. The balanced sets are ℂ itself, the empty set and the open and closed discs centered at 0 (visualizing complex numbers as points in the plane). Contrariwise, in the two dimensional Euclidean space there are many more balanced sets: any line segment with midpoint at (0,0) will do. As a result, ℂ and ℝ2 are entirely different as far as their vector space structure is concerned. * If p is a semi-norm on a linear space X, then for any constant c>0, the set is balanced. 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「Balanced set」の詳細全文を読む スポンサード リンク
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