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In mathematics, a Berkovich space, introduced by , is an analogue of an analytic space for ''p''-adic geometry, refining Tate's notion of a rigid analytic space. ==Berkovich spectrum== A seminorm on a ring ''A'' is a non-constant function ''f''→|''f''| from ''A'' to the non-negative reals such that |0| = 0, |1| = 1, |''f'' + ''g''| ≤ |''f''| + |''g''|, |''fg''| ≤ |''f''||''g''|. It is called multiplicative if |''fg''| = |''f''||''g''| and is called a norm if |''f''| = 0 implies ''f'' = 0. If ''A'' is a normed ring with norm ''f'' → ||''f''|| then the Berkovich spectrum of ''A'' is the set of multiplicative seminorms || on ''A'' that are bounded by the norm of ''A''. The Berkovich spectrum is topologized with the weakest topology such that for any ''f'' in ''A'' the map taking || to |''f''| is continuous.. The Berkovich spectrum of a normed ring ''A'' is non-empty if ''A'' is non-zero and is compact if ''A'' is complete. The spectral radius ρ(''f'') = lim |''f''''n''|1/''n'' of ''f'' is equal to sup''x''|''f''|''x'' 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「Berkovich space」の詳細全文を読む スポンサード リンク
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