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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 inverse theorem ： ウィキペディア英語版 Bounded inverse theorem In mathematics, the bounded inverse theorem (or inverse mapping theorem) is a result in the theory of bounded linear operators on Banach spaces. It states that a bijective bounded linear operator ''T'' from one Banach space to another has bounded inverse ''T''−1. It is equivalent to both the open mapping theorem and the closed graph theorem. It is necessary that the spaces in question be Banach spaces. For example, consider the space ''X'' of sequences ''x'' : N → R with only finitely many non-zero terms equipped with the supremum norm. The map ''T'' : ''X'' → ''X'' defined by :$T\; x\; =\; \backslash left(\; x\_,\; \backslash frac,\; \backslash frac,\; \backslash dots\; \backslash right)$ is bounded, linear and invertible, but ''T''−1 is unbounded. This does not contradict the bounded inverse theorem since ''X'' is not complete, and thus is not a Banach space. To see that it's not complete, consider the sequence of sequences ''x''(''n'') ∈ ''X'' given by :$x^\; =\; \backslash left(\; 1,\; \backslash frac1,\; \backslash dots,\; \backslash frac1,\; 0,\; 0,\; \backslash dots\; \backslash right)$ converges as ''n'' → ∞ to the sequence ''x''(∞) given by :$x^\; =\; \backslash left(\; 1,\; \backslash frac1,\; \backslash dots,\; \backslash frac1,\; \backslash dots\; \backslash right),$ which has all its terms non-zero, and so does not lie in ''X''. The completion of ''X'' is the space $c\_0$ of all sequences that converge to zero, which is a (closed) subspace of the ℓ''p'' space ℓ∞(N), which is the space of all bounded sequences. However, in this case, the map ''T'' is not onto, and thus not a bijection. To see this, one need simply note that the sequence :$x\; =\; \backslash left(\; 1,\; \backslash frac12,\; \backslash frac13,\; \backslash dots\; \backslash right),$ is an element of $c\_0$, but is not in the range of $T:c\_0\backslash to\; c\_0$. ==References== * (Section 8.2) 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 inverse theorem」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.