Editing Approximation property

{{Short description|Mathematical concept}}
{{about|the property in functional analysis|the notion in algebra (specifically ring theory)|approximation property (ring theory)}}
[[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  [[Stanisław Mazur]] (left) in 1936.<ref>[[Robert Megginson|Megginson, Robert E.]] ''An Introduction to Banach Space Theory'' p. 336</ref>]]
In [[mathematics]], specifically [[functional analysis]], 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.

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, much work in this area was done by [[Grothendieck]] (1955).

Later many other counterexamples were found. The space <math>\mathcal L(H)</math> of [[bounded operator]]s on an infinite-dimensional [[Hilbert space]] <math>H</math> does not have the approximation property.<ref>{{cite journal | url=https://dx.doi.org/10.1007/BF02392870 | doi=10.1007/BF02392870 | title=B(H) does not have the approximation propertydoes not have the approximation property | date=1981 | last1=Szankowski | first1=Andrzej | journal=Acta Mathematica | volume=147 | pages=89–108 }}</ref> 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.

== Definition ==
A [[locally convex]] topological vector space ''X'' is said to have '''the approximation property''', if the identity map can be approximated, uniformly on [[Relatively compact subspace|precompact set]]s, by continuous linear maps of finite rank.{{sfn | Schaefer|Wolff| 1999 | p=108-115}} 

For a locally convex space ''X'', the following are equivalent:{{sfn | Schaefer|Wolff| 1999 | p=108-115}}
# ''X'' has the approximation property;
# the closure of <math>X^{\prime} \otimes X</math> in <math>\operatorname{L}_p(X, X)</math> contains the identity map <math>\operatorname{Id} : X \to X</math>;
# <math>X^{\prime} \otimes X</math> is dense in <math>\operatorname{L}_p(X, X)</math>;
# for every locally convex space ''Y'', <math>X^{\prime} \otimes Y</math> is dense in <math>\operatorname{L}_p(X, Y)</math>;
# for every locally convex space ''Y'', <math>Y^{\prime} \otimes X</math> is dense in <math>\operatorname{L}_p(Y, X)</math>;
where <math>\operatorname{L}_p(X, Y)</math> denotes the space of continuous linear operators from ''X'' to ''Y'' endowed with the topology of uniform convergence on pre-compact subsets of ''X''. 

If ''X'' is a [[Banach space]] this requirement becomes that 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>.

== Related definitions ==
Some other flavours of the AP are studied:

Let <math>X</math> be a Banach space and let <math>1\leq\lambda<\infty</math>. We say that ''X'' 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>.

A Banach space is said to have '''bounded approximation property''' ('''BAP'''), if it has the <math>\lambda</math>-AP for some <math>\lambda</math>.

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.

== Examples ==

* Every subspace of an arbitrary product of Hilbert spaces possesses the approximation property.{{sfn | Schaefer|Wolff| 1999 | p=108-115}}  In particular,
** every Hilbert space has the approximation property. 
** every projective limit of Hilbert spaces, as well as any subspace of such a projective limit, possesses the approximation property.{{sfn | Schaefer|Wolff| 1999 | p=108-115}} 
** every [[nuclear space]] possesses the approximation property.
* Every separable Frechet space that contains a Schauder basis possesses the approximation property.{{sfn | Schaefer|Wolff| 1999 | p=108-115}} 
* 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 many spaces with the AP can be found. For example, the [[lp space|<math>\ell^p</math> spaces]], or the [[Tsirelson space|symmetric Tsirelson space]].

== References ==
{{Reflist}}

==Bibliography==
* {{cite journal|first=R. G.|last=Bartle|author-link=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}}
* [[Per Enflo|Enflo, P.]]: A counterexample to the approximation property in Banach spaces. ''Acta Math.'' 130, 309–317(1973).
* [[Grothendieck, A.]]: ''Produits tensoriels topologiques et espaces nucleaires''. Memo. Amer. Math. Soc. 16 (1955).
* {{cite journal|doi=10.2307/2321165|first=Paul R.|last=Halmos|author-link=Paul R. Halmos | title=Schauder bases | journal=[[American Mathematical Monthly]] |volume=85|year=1978|issue=4| pages=256–257 | jstor=2321165 |mr = 488901}}
* [[Paul R. Halmos]], "Has progress in mathematics slowed down?" ''Amer. Math. Monthly'' 97 (1990), no. 7, 561—588. {{MR|1066321}}
* William B. Johnson "Complementably universal separable Banach spaces" in [[Robert G. Bartle]] (ed.),  1980  ''Studies in functional analysis'',  Mathematical Association of America.
* 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.&nbsp;Centre de Math., École Polytech., Paris, 1973.  {{MR|407569}}
* [[Lindenstrauss, J.]]; Tzafriri, L.: Classical Banach Spaces I, Sequence spaces, 1977.
* {{cite journal|author1=Nedevski, P. |author2=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}}
* {{cite book|last=Pietsch|first=Albrecht|title=History of Banach spaces and linear operators|url=https://books.google.com/books?id=MMorKHumdZAC&dq=Pietsch&pg=PA203|publisher=Birkhäuser Boston, Inc.|location=Boston, MA|year=2007|pages=xxiv+855 pp|isbn=978-0-8176-4367-6|mr = 2300779}}
* [[Karen Saxe]], ''Beginning Functional Analysis'', [[Undergraduate Texts in Mathematics]], 2002 Springer-Verlag, New York.
* {{Cite book | isbn = 9780387987262 | title = Topological Vector Spaces | last1 = Schaefer | first1 = Helmut H. | authorlink = Helmut H. Schaefer | year = 1999 | publisher = [[Springer-Verlag]] | location = New York | last2 = Wolff | first2 = M.P. | series = [[Graduate Texts in Mathematics|GTM]] | volume = 3  }} <!-- Schaefer, H.H. (1999) Topological Vector Spaces -->
* Singer, Ivan. ''Bases in Banach spaces. II''. Editura Academiei Republicii Socialiste România, Bucharest; Springer-Verlag, Berlin-New York, 1981. viii+880 pp.&nbsp;{{ISBN|3-540-10394-5}}. {{MR|610799}}

{{Functional Analysis}}

[[Category:Operator theory]]
[[Category:Banach spaces]]