page_title: Approximation property
hierarchical_section_title: Approximation property
caption_attribution_description: Polski: Profesor Mazur wręcza żywą gęś szwedzkiemu matematykowi za rozwiązanie problemu z Księgi Szkockiej Fotografia pochodzi z książki Kazimierza Kuratowskiego "Pół wieku matematyki polskiej 1920-1970" wydanej przez Książkę i Wiedzę w 1973 r. (bez zastrzeżonych praw autorskich).
context_page_description: In mathematics, specifically functional analysis, a Banach space is said to have the approximation property, 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, much work in this area was done by Grothendieck.
  Later many other counterexamples were found. The space of bounded operators on does not have the approximation property. The spaces for and have closed subspaces that do not have the approximation property.
context_section_description: 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 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, much work in this area was done by Grothendieck (1955).
  Later many other counterexamples were found. The space of bounded operators on  does not have the approximation property (Szankowski). The spaces  for  and  (see Sequence space) have closed subspaces that do not have the approximation property.
caption: 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.[1]
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