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| [[Image:MazurGes.jpg|thumb|right|The construction of a Banach space without the approximation property earned [[Per Enflo]] a live goose in 1972, which had been promised by [[Stanislaw Mazur]] (left) in 1936.]] | | The person who wrote some sort of [https://www.flickr.com/search/?q=article article] is called Leland but it's not this most masucline name presently. Managing people is literally where his primary wage comes from. His wife and him live inside of Massachusetts and he has everything that he conditions there. Base jumping is something that he or she is been doing for months. He is running and maintaining a functional blog here: http://circuspartypanama.com<br><br>my site; [http://circuspartypanama.com clash of clans hack no survey no password] |
| In [[mathematics]], a [[Banach space]] is said to have the '''approximation property (AP)''', if every [[compact operator]] is a limit of [[finite-rank operator]]s. The converse is always true.
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| Every [[Hilbert space]] has this property. There are, however, [[Banach space]]s 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).
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| Later many other counterexamples were found. The space of [[bounded operator]]s on <math>\ell^2</math> does not have the approximation property ([[Szankowski]]). The spaces <math>\ell^p</math> for <math>p\neq 2</math> and <math>c_0</math> (see [[Sequence space]]) have closed subspaces that do not have the approximation property.
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| == Definition ==
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| A [[Banach space]] <math>X</math> is said to have the approximation property, if, for every [[compact set]] <math>K\subset X</math> and every <math>\varepsilon>0</math>, there is an [[operator (mathematics)|operator]] <math>T\colon X\to X</math> of finite rank so that <math>\|Tx-x\|\leq\varepsilon</math>, for every <math>x\in K</math>.
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| Some other flavours of the AP are studied:
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| Let <math>X</math> be a Banach space and let <math>1\leq\lambda<\infty</math>. We say that <math>X</math> has the <math>\lambda</math>''-approximation property'' (<math>\lambda</math>'''-AP'''), if, for every compact set <math>K\subset X</math> and every <math>\varepsilon>0</math>, there is an [[operator (mathematics)|operator]] <math>T\colon X\to X</math> of finite rank so that <math>\|Tx-x\|\leq\varepsilon</math>, for every <math>x\in K</math>, and <math>\|T\|\leq\lambda</math>.
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| A Banach space is said to have '''bounded approximation property''' ('''BAP'''), if it has the <math>\lambda</math>-AP for some <math>\lambda</math>.
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| A Banach space is said to have '''metric approximation property''' ('''MAP'''), if it is 1-AP.
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| A Banach space is said to have '''compact approximation property''' ('''CAP'''), if in the
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| definition of AP an operator of finite rank is replaced with a compact operator.
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| == Examples ==
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| Every space with a [[Schauder basis]] has the AP (we can use the projections associated to the base as the <math>T</math>'s in the definition), thus a lot of spaces with the AP can be found. For example, the [[lp space|<math>\ell^p</math> spaces]], or the [[Tsirelson space|symmetric Tsirelson space]].
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| == References ==
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| * {{cite journal|first=R. G.|last=Bartle|authorlink=Robert G. Bartle|title=MR0402468 (53 #6288) (Review of Per Enflo's "A counterexample to the approximation problem in Banach spaces" ''[[Acta Mathematica]]'' 130 (1973), 309–317)|journal=[[Mathematical Reviews]]|year=1977 | mr = 402468}}
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| * [[Per Enflo|Enflo, P.]]: A counterexample to the approximation property in Banach spaces. ''Acta Math.'' 130, 309–317(1973).
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| * [[Grothendieck, A.]]: ''Produits tensoriels topologiques et espaces nucleaires''. Memo. Amer. Math. Soc. 16 (1955).
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| * {{cite journal|doi=10.2307/2321165|first=Paul R.|last=Halmos|authorlink=Paul R. Halmos | title=Schauder bases | journal=[[American Mathematical Monthly]] |volume=85|year=1978|issue=4| pages=256–257 | jstor=2321165 |mr = 488901}}
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| * [[Paul R. Halmos]], "Has progress in mathematics slowed down?" ''Amer. Math. Monthly'' 97 (1990), no. 7, 561—588. {{MR|1066321}}
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| * William B. Johnson "Complementably universal separable Banach spaces" in [[Robert G. Bartle]] (ed.), 1980 ''Studies in functional analysis'', Mathematical Association of America.
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| * Kwapień, S. "On Enflo's example of a Banach space without the approximation property". Séminaire Goulaouic–Schwartz 1972—1973: Équations aux dérivées partielles et analyse fonctionnelle, Exp. No. 8, 9 pp. Centre de Math., École Polytech., Paris, 1973. {{MR|407569}}
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| * [[Lindenstrauss, J.]]; Tzafriri, L.: Classical Banach Spaces I, Sequence spaces, 1977.
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| * {{cite journal|author=Nedevski, P.; Trojanski, S. |title=P. Enflo solved in the negative Banach's problem on the existence of a basis for every separable Banach space | journal=Fiz.-Mat. Spis. Bulgar. Akad. Nauk. |volume=16 |issue=49 |year=1973 |pages=134–138 | mr = 458132}}
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| * {{cite book|last=Pietsch|first=Albrecht|title=History of Banach spaces and linear operators|url=http://books.google.com/books?id=MMorKHumdZAC&pg=PA203&dq=Pietsch|publisher=Birkhäuser Boston, Inc.|location=Boston, MA|year=2007|pages=xxiv+855 pp.|isbn=978-0-8176-4367-6|mr = 2300779}}
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| * Karen Saxe, ''Beginning Functional Analysis'', Undergraduate Texts in Mathematics, 2002 Springer-Verlag, New York.
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| * Singer, Ivan. ''Bases in Banach spaces. II''. Editura Academiei Republicii Socialiste România, Bucharest; Springer-Verlag, Berlin-New York, 1981. viii+880 pp. ISBN 3-540-10394-5. {{MR|610799}}
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| [[Category:Operator theory]]
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| [[Category:Banach spaces]]
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| [[Category:Functional analysis]]
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