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. There are, however, Banach spaces which do not; Enflo published the first counterexample in an 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.

Definition

A Banach space $X$ is said to have the approximation property, if, for every compact set $Ksubset\; X$ and every $varepsilon>0$, there is an operator $Tcolon\; X\; o\; X$ of finite rank so that $|Tx-x|leqvarepsilon$, for every $xin\; K$.

Some other flavours of the AP are studied:

Let $X$ be a Banach space and let -AP), if, for every compact set $Ksubset\; X$ and every $varepsilon>0$, there is an operator $Tcolon\; X\; o\; X$ of finite rank so that $|Tx-x|leqvarepsilon$, for every $xin\; K$, and $|T|leqlambda$.

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.