APPROXIMATION PROPERTY

In mathematics, a Banach space is said to have the approximation property ('AP' in short), if every compact operator is a limit of finite rank operators. The converse is always true.
Every Hilbert space has this property; for a general Banach space, this was unknown till Enflo's 1973 article. However, a lot of work in this area was done by Grothendieck (1955).
Later many other counterexamples were found. The space of bounded operators on $l\_2$ does not have the approximation property (Shankovskii). The spaces $l\_p$ for $p$
ot =2 and $c\_0$ (see Sequence space) have closed subspaces that do not have the approximation property.

Contents

Definition

Examples

References

Definition

A Banach space $X$ is said to have the approximation property, if, for every compact set $Ksubset\; X$ and every , there is an operator $Tcolon\; X\; o\; X$ of finite rank so that , for every $xin\; K$.
Some other flavours of the AP are studied:
Let $X$ be a Banach space and let
A Banach space is said to have 'bounded approximation property' ('BAP'), if it has the $lambda$-AP for some $lambda$.
A Banach space is said to have 'metric approximation property' ('MAP'), if it is 1-AP.
A Banach space is said to have 'compact approximation property' ('CAP'), if in the
definition of AP an operator of finite rank is replaced with a compact operator.

Examples

Every space with a Schauder basis has the AP (we can use the projections associated to the base as the $T$'s in the definition), thus a lot of spaces with the AP can be found. For example, the l^''p'' spaces, or the symmetric Tsirelson space.

References

★ Enflo, P.: A counterexample to the approximation property in Banach spaces. 'Acta Math.' 130, 309–317(1973).

★ Grothendieck, A.: Produits tensoriels topologiques et espaces nucleaires. Memo. Amer. Math. Soc. 16 (1955).

★ Lindenstrauss, J.; Tzafriri, L.: Classical Banach Spaces I, Sequence spaces, 1977.