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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Cauchy-Schwarz inequality ： ウィキペディア英語版 Cauchy–Schwarz inequality In mathematics, the Cauchy–Schwarz inequality is a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, and other areas. It is considered to be one of the most important inequalities in all of mathematics.〔(The Cauchy–Schwarz Master Class: an Introduction to the Art of Mathematical Inequalities, Ch. 1 ) by J. Michael Steele.〕 It has a number of generalizations, among them Hölder's inequality. The inequality for sums was published by , while the corresponding inequality for integrals was first proved by . The modern proof of the integral inequality was given by .〔 == Statement of the inequality == The Cauchy–Schwarz inequality states that for all vectors ''x'' and ''y'' of an inner product space it is true that :$|\backslash langle\; x,y\backslash rangle|\; ^2\; \backslash leq\; \backslash langle\; x,x\backslash rangle\; \backslash cdot\; \backslash langle\; y,y\backslash rangle,$ where $\backslash langle\backslash cdot,\backslash cdot\backslash rangle$ is the inner product also known as dot product. Equivalently, by taking the square root of both sides, and referring to the norms of the vectors, the inequality is written as : $|\backslash langle\; x,y\backslash rangle|\; \backslash leq\; \backslash |x\backslash |\; \backslash cdot\; \backslash |y\backslash |.\backslash ,$ Moreover, the two sides are equal if and only if ''x'' and ''y'' are linearly dependent (or, in a geometrical sense, they are parallel or one of the vector's magnitudes is zero). If $x\_1,\backslash ldots,\; x\_n\backslash in\backslash mathbb\; C$ and $y\_1,\backslash ldots,\; y\_n\backslash in\backslash mathbb\; C$ have an imaginary component, the inner product is the standard inner product and the bar notation is used for complex conjugation then the inequality may be restated more explicitly as :$|x\_1\; \backslash bar\_1\; +\; \backslash cdots\; +\; x\_n\; \backslash bar\_n|^2\; \backslash leq\; (|x\_1|^2\; +\; \backslash cdots\; +\; |x\_n|^2)\; (|y\_1|^2\; +\; \backslash cdots\; +\; |y\_n|^2).$ When viewed in this way the numbers ''x''1, ..., ''x''''n'', and ''y''1, ..., ''y''''n'' are the components of ''x'' and ''y'' with respect to an orthonormal basis of ''V''. Even more compactly written: :$\backslash left|\; \backslash sum\_^n\; x\_i\; \backslash bar\_i\; \backslash right|^2\; \backslash leq\; \backslash sum\_^n\; |x\_j|^2\; \backslash sum\_^n\; |y\_k|^2\; .$ Equality holds if and only if ''x'' and ''y'' are linearly dependent, that is, one is a scalar multiple of the other (which includes the case when one or both are zero). The finite-dimensional case of this inequality for real vectors was proven by Cauchy in 1821, and in 1859 Cauchy's student Bunyakovsky noted that by taking limits one can obtain an integral form of Cauchy's inequality. The general result for an inner product space was obtained by Schwarz in the year 1888. 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「Cauchy–Schwarz inequality」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.