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In mathematical analysis Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of . :Theorem (Hölder's inequality). Let be a measure space and let with . Then, for all measurable real- or complex-valued functions and on , :: :If, in addition, and and , then Hölder's inequality becomes an equality if and only if and are linearly dependent in , meaning that there exist real numbers , not both of them zero, such that -almost everywhere. The numbers and above are said to be Hölder conjugates of each other. The special case gives a form of the Cauchy–Schwarz inequality. Hölder's inequality holds even if is infinite, the right-hand side also being infinite in that case. Conversely, if is in and is in , then the pointwise product is in . Hölder's inequality is used to prove the Minkowski inequality, which is the triangle inequality in the space , and also to establish that is the dual space of for . Hölder's inequality was first found by , and discovered independently by . ==Remarks== 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「Hölder's inequality」の詳細全文を読む スポンサード リンク
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