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In mathematics, an ultrametric space is a special kind of metric space in which the triangle inequality is replaced with . Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Although some of the theorems for ultrametric spaces may seem strange at a first glance, they appear naturally in many applications. == Formal definition == Formally, an ultrametric space is a set of points with an associated distance function (also called a metric) : (where is the set of real numbers), such that for all , one has: # # iff # (symmetry) # (strong triangle or ultrametric inequality). In the case when is a group and is generated by a length function (so that ), the last property can be made stronger using the Krull sharpening〔Planet Math: (Ultrametric Triangle Inequality )〕 to: : with equality if . We want to prove that if , then the equality occurs if . Without loss of generality, let us assume that . This implies that . But we can also compute . Now, the value of cannot be , for if that is the case, we have contrary to the initial assumption. Thus, , and . Using the initial inequality, we have and therefore . 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「ultrametric space」の詳細全文を読む スポンサード リンク
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