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.

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.[1]

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 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 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.[2]

For a locally convex space X, the following are equivalent:[2]

1. X has the approximation property;
2. the closure of ${\displaystyle X^{\prime }\otimes X}$ in ${\displaystyle \operatorname {L} _{p}(X,X)}$ contains the identity map ${\displaystyle \operatorname {Id} :X\to X}$;
3. ${\displaystyle X^{\prime }\otimes X}$ is dense in ${\displaystyle \operatorname {L} _{p}(X,X)}$;
4. for every locally convex space Y, ${\displaystyle X^{\prime }\otimes Y}$ is dense in ${\displaystyle \operatorname {L} _{p}(X,Y)}$;
5. for every locally convex space Y, ${\displaystyle Y^{\prime }\otimes X}$ is dense in ${\displaystyle \operatorname {L} _{p}(Y,X)}$;

where ${\displaystyle \operatorname {L} _{p}(X,Y)}$ 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 ${\displaystyle K\subset X}$ 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 ${\displaystyle x\in K}$.

Related definitions

Some other flavours of the AP are studied:

Let ${\displaystyle X}$ be a Banach space and let ${\displaystyle 1\leq \lambda <\infty }$. We say that X has the ${\displaystyle \lambda }$-approximation property (${\displaystyle \lambda }$-AP), if, 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 ${\displaystyle x\in K}$, and ${\displaystyle \|T\|\leq \lambda }$.

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.

Examples

• Every subspace of an arbitrary product of Hilbert spaces possesses the approximation property.[2] 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.[2]
  • every nuclear space possesses the approximation property.
• Every separable Frechet space that contains a Schauder basis possesses the approximation property.[2]
• Every space with a Schauder basis has the AP (we can use the projections associated to the base as the ${\displaystyle T}$'s in the definition), thus many spaces with the AP can be found. For example, the ${\displaystyle \ell ^{p}}$ spaces, or the symmetric Tsirelson space.

References

1. Megginson, Robert E. An Introduction to Banach Space Theory p. 336
2. Schaefer & Wolff 1999, p. 108-115.

Bibliography

• Bartle, R. G. (1977). "MR0402468 (53 #6288) (Review of Per Enflo's "A counterexample to the approximation problem in Banach spaces" Acta Mathematica 130 (1973), 309–317)". Mathematical Reviews. MR 0402468.
• 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).
• Halmos, Paul R. (1978). "Schauder bases". American Mathematical Monthly. 85 (4): 256–257. doi:10.2307/2321165. JSTOR 2321165. MR 0488901.
• Paul R. Halmos, "Has progress in mathematics slowed down?" Amer. Math. Monthly 97 (1990), no. 7, 561—588. MR1066321
• 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. MR407569
• Lindenstrauss, J.; Tzafriri, L.: Classical Banach Spaces I, Sequence spaces, 1977.
• Nedevski, P.; Trojanski, S. (1973). "P. Enflo solved in the negative Banach's problem on the existence of a basis for every separable Banach space". Fiz.-Mat. Spis. Bulgar. Akad. Nauk. 16 (49): 134–138. MR 0458132.
• Pietsch, Albrecht (2007). History of Banach spaces and linear operators. Boston, MA: Birkhäuser Boston, Inc. pp. xxiv+855 pp. ISBN 978-0-8176-4367-6. MR 2300779.
• Karen Saxe, Beginning Functional Analysis, Undergraduate Texts in Mathematics, 2002 Springer-Verlag, New York.
• Schaefer, Helmuth H.; Wolff, M.P. (1999). Topological Vector Spaces. GTM. 3. New York: Springer-Verlag. ISBN 9780387987262.
• 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. MR610799