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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
, a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
is said to have the approximation property (AP), if every
compact operator In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact c ...
is a limit of finite-rank operators. The converse is always true. Every
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
has this property. There are, however,
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s which do not;
Per Enflo Per H. Enflo (; born 20 May 1944) is a Swedish mathematician working primarily in functional analysis, a field in which he solved mathematical problems, problems that had been considered fundamental. Three of these problems had been open problem, ...
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 operator In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector ...
s on \ell^2 does not have the approximation property.Szankowski, A.
B(H) does not have the approximation property.
''Acta Math.'' 147, 89-108(1981).
The spaces \ell^p for p\neq 2 and c_0 (see
Sequence space In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural nu ...
) have closed subspaces that do not have the approximation property.


Definition

A
locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
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 In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
this requirement becomes that for every
compact set In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i. ...
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 In mathematics, a Schauder basis or countable basis is similar to the usual ( Hamel) basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums. This ...
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 Karen Saxe is an American mathematician who specializes in functional analysis, and in the mathematical study of issues related to social justice. She is DeWitt Wallace Professor of Mathematics, Emerita at Macalester College,. She is Associate Exec ...
, ''Beginning Functional Analysis'',
Undergraduate Texts in Mathematics Undergraduate Texts in Mathematics (UTM) (ISSN 0172-6056) is a series of undergraduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are small yellow bo ...
, 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