# Approximation property

The construction of a Banach space without the approximation property earned Per Enflo a live goose in 1972, which had been promised by Stanisław Mazur (left) in 1936.

In mathematics, 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, a lot of work in this area was done by Grothendieck (1955).

Later many other counterexamples were found. The space of bounded operators on ${\displaystyle \ell ^{2}}$ does not have the approximation property (Szankowski). 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.

## Definition

A locally convex topological vector space ${\displaystyle X}$ is said to have the approximation property, if the identity map can be approximated, uniformly on precompact sets, by continuous linear maps of finite rank.[1] If X is a Banach space the this requirement becomes that for every compact set ${\displaystyle K\subset X}$ and every ${\displaystyle \varepsilon >0}$, there is an operator ${\displaystyle T\colon X\to X}$ of finite rank so that ${\displaystyle \|Tx-x\|\leq \varepsilon }$, for every ${\displaystyle x\in K}$.

Some other flavours of the AP are studied:

A Banach space is said to have bounded approximation property (BAP), if it has the ${\displaystyle \lambda }$-AP for some ${\displaystyle \lambda }$.

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.

## References

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• 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).
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• Paul R. Halmos, "Has progress in mathematics slowed down?" Amer. Math. Monthly 97 (1990), no. 7, 561—588. Template:MR
• 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. Template:MR
• Lindenstrauss, J.; Tzafriri, L.: Classical Banach Spaces I, Sequence spaces, 1977.
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• Karen Saxe, Beginning Functional Analysis, Undergraduate Texts in Mathematics, 2002 Springer-Verlag, New York.
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• Singer, Ivan. Bases in Banach spaces. II. Editura Academiei Republicii Socialiste România, Bucharest; Springer-Verlag, Berlin-New York, 1981. viii+880 pp. ISBN 3-540-10394-5. Template:MR
1. Schaefer p. 108
2. Schaefer p. 110
3. Schaefer p. 109
4. Schaefer p. 115