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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 dual norm ： ウィキペディア英語版 dual norm The concept of a dual norm arises in functional analysis, a branch of mathematics. Let $X$ be a normed space (or, in a special case, a Banach space) over a number field $F$ (i.e. $F=$ or $F=$) with norm $\backslash left\backslash |\; \backslash cdot\; \backslash right\backslash |$. Then the ''dual'' (or ''conjugate'') ''normed'' space $X\text{'}$ (another notation $X^$ *) is defined as the set of all continuous linear functionals from $X$ into the base field $F$. If $f:X\backslash to\; F$ is such a linear functional, then the ''dual norm'' $\backslash |\backslash cdot\backslash |\text{'}$ of $f$ is defined by : $\backslash left\backslash |\; f\; \backslash right\backslash |\; \text{'}\; =\; \backslash sup\backslash \; =\; \backslash sup\; \backslash left\backslash :\; x\; \backslash in\; X,\; x\; \backslash ne\; 0\; \backslash right\backslash \}\; .$ With this norm, the dual space $X\text{'}$ is also a normed space, and moreover a Banach space, since $X\text{'}$ is always complete.〔http://www.seas.ucla.edu/~vandenbe/236C/lectures/proxop.pdf〕 == Examples == # Dual Norm of Vectors #:If ''p'', ''q'' ∈ $(\backslash infty)$ satisfy $1/p+1/q=1$, then the ℓ''p'' and ℓ''q'' norms are dual to each other. #:In particular the Euclidean norm is self-dual (). Similarly, the Schatten ''p''-norm on matrices is dual to the Schatten ''q''-norm. #:For $\backslash sqrt$, the dual norm is $\backslash sqrty\}$ with $Q$ positive definite. # Dual Norm of Matrices #:''Frobenius norm'' #::$\backslash left\backslash |\; A\; \backslash right\backslash |\; \_^m\backslash sum\_^n\; \backslash left|\; a\_\; \backslash right|\; ^2\}\; =\; \backslash sqrt^2\}$ #:Its dual norm is $\backslash left\backslash |\; B\; \backslash right\backslash |\; \_(A)$ #:Dual norm $\backslash sum\_i\; \backslash sigma\_i(B)$ 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「dual norm」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.