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 $\ell^2$ does not have the approximation property (Szankowski). The spaces $\ell^p$ for $p\neq 2$ and $c_0$ (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 sets, by continuous linear maps of finite rank. If X is a Banach space the this requirement becomes that for every compact set $K\subset X$ and every $\varepsilon>0$, there is an operator $T\colon X\to X$ of finite rank so that $\|Tx-x\|\leq\varepsilon$, for every $x \in K$.

Some other flavours of the AP are studied:

Let $X$ be a Banach space and let $1\leq\lambda<\infty$. We say that X has the $\lambda$-approximation property ($\lambda$-AP), if, for every compact set $K\subset X$ and every $\varepsilon>0$, there is an operator $T\colon X \to X$ of finite rank so that $\|Tx - x\|\leq\varepsilon$, for every $x \in K$, and $\|T\|\leq\lambda$.

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 projective limit of Hilbert spaces, as well as any subspace of such a projective limit, possesses the approximation property.
  • Hence every nuclear space possesses the approximation property.
• Every subspace of an arbitrary product of Hilbert spaces possesses the approximation property.
• Every separable Frechet space that contains a Schauder basis possesses the approximation property.
• 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 $\ell^p$ 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).
• Paul R. Halmos, "Has progress in mathematics slowed down?" Amer. Math. Monthly 97 (1990), no. 7, 561—588. MR 1066321
• 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. MR 407569
• 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. ISBN 3-540-10394-5. MR 610799