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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 Banach limit ： ウィキペディア英語版 Banach limit In mathematical analysis, a Banach limit is a continuous linear functional $\backslash phi:\; \backslash ell^\backslash infty\; \backslash to\; \backslash mathbb$ defined on the Banach space $\backslash ell^\backslash infty$ of all bounded complex-valued sequences such that for all sequences $x=(x\_n)$, $y=(y\_n)$ in $\backslash ell^\backslash infty$, and complex numbers $\backslash alpha$: # $\backslash phi(\backslash alpha\; x+y)=\backslash alpha\backslash phi(x)+\backslash phi(y)$ (linearity); # if $x\_n\backslash geq\; 0$ for all $n\backslash in\; \backslash mathbb$, then $\backslash phi(x)\backslash geq\; 0$ (positivity); # $\backslash phi(x)=\backslash phi(Sx)$, where $S$ is the shift operator defined by $(Sx)\_n=x\_$ (shift-invariance); # if $x$ is a convergent sequence, then $\backslash phi(x)=\backslash lim\; x$. Hence, $\backslash phi$ is an extension of the continuous functional $\backslash lim\; x:c\backslash mapsto\; \backslash mathbb\; C$ where $c\; \backslash subset\backslash ell^\backslash infty$ is the complex vector space of all sequences with converge to a (usual) limit in $\backslash mathbb\; C$. In other words, a Banach limit extends the usual limits, is linear, shift-invariant and positive. However, there exist sequences for which the values of two Banach limits do not agree. We say that the Banach limit is not uniquely determined in this case. As a consequence of the above properties, a Banach limit also satisfies: : $\backslash liminf\_\; x\_n\backslash le\backslash phi(x)\; \backslash le\; \backslash limsup\_x\_n$ The existence of Banach limits is usually proved using the Hahn–Banach theorem (analyst's approach), or using ultrafilters (this approach is more frequent in set-theoretical expositions). These proofs necessarily use the Axiom of choice (so called non-effective proof). ==Almost convergence== There are non-convergent sequences which have a uniquely determined Banach limit. For example, if $x=(1,0,1,0,\backslash ldots)$, then $x+S(x)=(1,1,1,\backslash ldots)$ is a constant sequence, and :$2\backslash phi(x)=\backslash phi(x)+\backslash phi(Sx)=\backslash phi(x+Sx)=\backslash phi((1,1,1,\backslash ldots))=\backslash lim((1,1,1,\backslash ldots))=1$ holds. Thus, for any Banach limit, this sequence has limit $1/2$. A bounded sequence $x$ with the property, that for every Banach limit $\backslash phi$ the value $\backslash phi(x)$ is the same, is called almost convergent. 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「Banach limit」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.