Approximation property

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 ${\displaystyle {\mathcal {L}}(H)}$ of bounded operators on an infinite-dimensional Hilbert space ${\displaystyle H}$ does not have the approximation property.^[2] The spaces ${\displaystyle \ell ^{p}}$ for ${\displaystyle p\neq 2}$ and ${\displaystyle c_{0}}$ (see Sequence space) have closed subspaces that do not have the approximation property.