Approximation property

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).

Later many other counterexamples were found. The space $\mathcal L(H)$ of bounded operators on an infinite-dimensional Hilbert space $H$ does not have the approximation property. 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.

For a locally convex space ''X'', the following are equivalent:
# ''X'' has the approximation property;
# the closure of $X^ \otimes X$ in $\operatorname_p(X, X)$ contains the identity map $\operatorname : X \to X$;
# $X^ \otimes X$ is dense in $\operatorname_p(X, X)$;
# for every locally convex space ''Y'', $X^ \otimes Y$ is dense in $\operatorname_p(X, Y)$;
# for every locally convex space ''Y'', $Y^ \otimes X$ is dense in $\operatorname_p(Y, X)$;
where $\operatorname_p(X, Y)$ 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 $K\subset X$ and every $\varepsilon>0$, there is an operator $T\colon X\to X$ of finite rank so that $\, Tx-x\, \leq\varepsilon$, for every $x \in K$.

Related definitions

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 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 has the AP (we can use the projections associated to the base as the $T$'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. .

{{Functional Analysis

Operator theory
Banach spaces