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 $$X be a Banach space and let $$1leqlambda. We say that $$X has the $$lambda"-approximation property" ($$lambda-AP), if, for every compact set $$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.