Approximation property
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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 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
External links
- Approximation property @ Wikipedia
