Editing Approximation property (section)

{{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.