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).

of bounded operators on an infinite-dimensional Hilbert space does not have the approximation property. (see Sequence space) have closed subspaces that do not have the approximation property.

Definition

A locally convex topological vector space X is said to have the approximation property, if the identity map can be approximated, uniformly on precompact setss, by continuous linear maps of finite rank.

For a locally convex space X, the following are equivalent:

1. X has the approximation property;

2. ;

3. ;

4. for every locally convex space Y;

5. for every locally convex space Y;

denotes the space of continuous linear operators from X to Y endowed with the topology of uniform convergence on pre-compact subsets of X.

If X is a Banach space this requirement becomes that for every compact set, there is an operator.

Related definitions

Some other flavours of the AP are studied:

. We say that X-approximation property-AP, there is an operator.

A Banach space is said to have bounded approximation property (BAP.

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 subspace of an arbitrary product of Hilbert spaces possesses the approximation property. In particular,

• every Hilbert space has the approximation property.

• every projective limit of Hilbert spaces, as well as any subspace of such a projective limit, possesses the approximation property.

• every nuclear space possesses the approximation property.

• Every separable Frechet space that contains a Schauder basis possesses the approximation property.

• Every space with a Schauder basis's in the definition), thus many spaces with the AP can be found. For example, the \ell^p spaces, or the symmetric Tsirelson space.

References

Bibliography

• 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).

• Paul R. Halmos, "Has progress in mathematics slowed down?" Amer. Math. Monthly 97 (1990), no. 7, 561–588.

• William B. Johnson "Complementably universal separable Banach spaces" in Robert G. Bartle (ed.), 1980 Studies in functional analysis, Mathematical Association of America.

• Kwapień, S. "On Enflo's example of a Banach space without the approximation property". Séminaire Goulaouic–Schwartz 1972—1973: Équations aux dérivées partielles et analyse fonctionnelle, Exp. No. 8, 9 pp.Centre de Math., École Polytech., Paris, 1973.

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

• Karen Saxe, Beginning Functional Analysis, Undergraduate Texts in Mathematics, 2002 Springer-Verlag, New York.

• Singer, Ivan. Bases in Banach spaces. II. Editura Academiei Republicii Socialiste România, Bucharest; Springer-Verlag, Berlin-New York, 1981. viii+880 pp..