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 ${\displaystyle {\mathcal {L}}(H)}${\displaystyle {\mathcal {L}}(H)} of bounded operators on an infinite-dimensional Hilbert space ${\displaystyle H}${\displaystyle H} does not have the approximation property.^[2] The spaces ${\displaystyle \ell ^{p}}${\displaystyle \ell ^{p}} for ${\displaystyle p\neq 2}${\displaystyle p\neq 2} and ${\displaystyle c_{0}}${\displaystyle c_{0}} (see Sequence space) have closed subspaces that do not have the approximation property.

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.^[3]

For a locally convex space X, the following are equivalent:^[3]

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

where ${\displaystyle \operatorname {L} _{p}(X,Y)}${\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 ${\displaystyle K\subset X}${\displaystyle K\subset X} and every ${\displaystyle \varepsilon >0}${\displaystyle \varepsilon >0}, there is an operator ${\displaystyle T\colon X\to X}${\displaystyle T\colon X\to X} of finite rank so that ${\displaystyle \|Tx-x\|\leq \varepsilon }${\displaystyle \|Tx-x\|\leq \varepsilon }, for every ${\displaystyle x\in K}${\displaystyle x\in K}.

Some other flavours of the AP are studied:

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

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

• Every subspace of an arbitrary product of Hilbert spaces possesses the approximation property.^[3] 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.^[3]
  • every nuclear space possesses the approximation property.
• Every separable Frechet space that contains a Schauder basis possesses the approximation property.^[3]
• Every space with a Schauder basis has the AP (we can use the projections associated to the base as the ${\displaystyle T}${\displaystyle T}'s in the definition), thus many spaces with the AP can be found. For example, the ${\displaystyle \ell ^{p}}${\displaystyle \ell ^{p}} spaces, or the symmetric Tsirelson space.

• 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. 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 0407569
• 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, Helmut H.; Wolff, M.P. (1999). Topological Vector Spaces. GTM. Vol. 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. MR 0610799

• Operator theory
• Banach spaces

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