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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 Approximation property ： ウィキペディア英語版 Approximation property In mathematics, a Banach space is said to have the approximation property (AP), if every compact operator is a limit of finite-rank operators. The converse is always true. Every Hilbert space has this property. There are, however, Banach spaces which do not; Per Enflo published the first counterexample in a 1973 article. However, a lot of work in this area was done by Grothendieck (1955). Later many other counterexamples were found. The space of bounded operators on $\backslash ell^2$ does not have the approximation property (Szankowski). The spaces $\backslash ell^p$ for $p\backslash neq\; 2$ and $c\_0$ (see Sequence space) have closed subspaces that do not have the approximation property. == Definition == A locally convex topological vector space $X$ is said to have the approximation property, if the identity map can be approximated, uniformly on precompact sets, by continuous linear maps of finite rank.〔Schaefer p. 108〕 If ''X'' is a Banach space the this requirement becomes that for every compact set $K\backslash subset\; X$ and every $\backslash varepsilon>0$, there is an operator $T\backslash colon\; X\backslash to\; X$ of finite rank so that $\backslash |Tx-x\backslash |\backslash leq\backslash varepsilon$, for every $x\; \backslash in\; K$. Some other flavours of the AP are studied: Let $X$ be a Banach space and let . We say that ''X'' has the $\backslash lambda$''-approximation property'' ($\backslash lambda$-AP), if, for every compact set $K\backslash subset\; X$ and every $\backslash varepsilon>0$, there is an operator $T\backslash colon\; X\; \backslash to\; X$ of finite rank so that $\backslash |Tx\; -\; x\backslash |\backslash leq\backslash varepsilon$, for every $x\; \backslash in\; K$, and $\backslash |T\backslash |\backslash leq\backslash lambda$. A Banach space is said to have bounded approximation property (BAP), if it has the $\backslash lambda$-AP for some $\backslash lambda$. A Banach space is said to have metric approximation property (MAP), if it is 1-AP. A Banach space is said to have compact approximation property (CAP), if in the definition of AP an operator of finite rank is replaced with a compact operator. 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「Approximation property」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.