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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 \ell^2 does not have the approximation property (Szankowski). The spaces \ell^p for p\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\subset X and every \varepsilon>0, there is an operator T\colon X\to X of finite rank so that \|Tx-x\|\leq\varepsilon, for every x \in K.
Some other flavours of the AP are studied:
Let X be a Banach space and let 1\leq\lambda<\infty. We say that ''X'' has the \lambda''-approximation property'' (\lambda-AP), if, for every compact set K\subset X and every \varepsilon>0, there is an operator T\colon X \to X of finite rank so that \|Tx - x\|\leq\varepsilon, for every x \in K, and \|T\|\leq\lambda.
A Banach space is said to have bounded approximation property (BAP), if it has the \lambda-AP for some \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)
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