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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 Closed range theorem ： ウィキペディア英語版 Closed range theorem In the mathematical theory of Banach spaces, the closed range theorem gives necessary and sufficient conditions for a closed densely defined operator to have closed range. ==History== The theorem was proved by Stefan Banach in his 1932 ''Théorie des opérations linéaires''. ==Statement== Let $X$ and $Y$ be Banach spaces, $T\backslash colon\; D(T)\; \backslash to\; Y$ a closed linear operator whose domain $D(T)$ is dense in $X$, and $T\text{'}$ the transpose of $T$. The theorem asserts that the following conditions are equivalent: * $R(T)$, the range of $T$, is closed in $Y$, * $R(T\text{'})$, the range of $T\text{'}$, is closed in $X\text{'}$, the dual of $X$, * $R(T)\; =\; N(T\text{'})^\backslash perp=\backslash$, * $R(T\text{'})\; =\; N(T)^\backslash perp=\backslash$. 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「Closed range 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.