Karl Jones: Created page with "In mathematics, a Banach space is said to have the '''approximation property''' (AP), if every compact operator is a limit of Finite-rank operator|finite-rank op..."

2016-09-21T20:49:12Z

Created page with "In mathematics, a Banach space is said to have the '''approximation property''' (AP), if every compact operator is a limit of Finite-rank operator|finite-rank op..."

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In [[mathematics]], a [[Banach space]] is said to have the '''approximation property''' (AP), if every [[compact operator]] is a limit of [[Finite-rank operator|finite-rank operators]].

The converse is always true.

== Description ==

Every [[Hilbert space]] has this property.

There are, however, Banach spaces which do not; [[Per Enflo]] published the first counterexample in a 1973 article. (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]] in 1936.) Much work in this area was also done by Grothendieck (1955).

Later many other counterexamples were found.

== See also ==

* [[Banach space]]

== External links ==

* [https://en.wikipedia.org/wiki/Approximation_property Approximation property] @ Wikipedia

[[Category:Mathematics]]

Approximation property - Revision history

Karl Jones: Created page with "In mathematics, a Banach space is said to have the '''approximation property''' (AP), if every compact operator is a limit of Finite-rank operator|finite-rank op..."