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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 Bounded operator ： ウィキペディア英語版 Bounded operator In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation ''L'' between normed vector spaces ''X'' and ''Y'' for which the ratio of the norm of ''L''(''v'') to that of ''v'' is bounded by the same number, over all non-zero vectors ''v'' in ''X''. In other words, there exists some ''M'' > 0 such that for all ''v'' in ''X'' :$\backslash |Lv\backslash |\_Y\; \backslash le\; M\; \backslash |v\backslash |\_X.\backslash ,\; \backslash ,$ The smallest such ''M'' is called the operator norm $\backslash |L\backslash |\_\{\backslash mathrm\{op\}\}\; \backslash ,$ of ''L''. A bounded linear operator is generally not a bounded function; the latter would require that the norm of ''L''(''v'') be bounded for all ''v'', which is not possible unless ''Y'' is the zero vector space. Rather, a bounded linear operator is a locally bounded function. A linear operator between normed spaces is bounded if and only if it is continuous, and by linearity, if and only if it is continuous at zero. ==Examples== * Any linear operator between two finite-dimensional normed spaces is bounded, and such an operator may be viewed as multiplication by some fixed matrix. * Many b )\times (d )\to \, :is a continuous function, then the operator $L,\; \backslash ,$ defined on the space $C(b)\; \backslash ,$ of continuous functions on $(b)\; \backslash ,$ endowed with the uniform norm and with values in the space $C(d),\; \backslash ,$ with $L\; \backslash ,$ given by the formula ::$(Lf)(y)=\backslash int\_^\backslash !K(x,\; y)f(x)\backslash ,dx,\; \backslash ,$ :is bounded. This operator is in fact compact operator" TITLE="integral transforms are bounded linear operators. For instance, if ::$K:(b)\backslash times\; (d)\backslash to\; \backslash ,$ :is a continuous function, then the operator $L,\; \backslash ,$ defined on the space $C(b)\; \backslash ,$ of continuous functions on $(b)\; \backslash ,$ endowed with the uniform norm and with values in the space $C(d),\; \backslash ,$ with $L\; \backslash ,$ given by the formula ::$(Lf)(y)=\backslash int\_^\backslash !K(x,\; y)f(x)\backslash ,dx,\; \backslash ,$ :is bounded. This operator is in fact compact operator">compact. The compact operators form an important class of bounded operators. * The Laplace operator ::$\backslash Delta:H^2(^n)\backslash to\; L^2(^n)\; \backslash ,$ :(its domain of a function values in a space of square integrable functions) is bounded. * The shift operator on the ''l''''2'' space of all sequences (''x''''0'', ''x''''1'', ''x''''2''...) of real numbers with ::$L(x\_0,\; x\_1,\; x\_2,\; \backslash dots)=(0,\; x\_0,\; x\_1,\; x\_2,\backslash dots)\; \backslash ,$ :is bounded. Its operator norm is easily seen to be 1. 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「Bounded operator」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.