Approximation property
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In mathematics, a Banach space is said to have the approximation property (AP in short), if every compact operator is a limit of finite rank operators. The converse is always true.
Every Hilbert space has this property; for a general Banach space, this was unknown till Enflo's 1973 article. However, a lot of work in this area was done by Grothendieck (1955).
Definition
A Banach space <math>X<math> is said to have the approximation property, if, for every compact set <math>K\subset X<math> and every <math>\varepsilon>0<math>, there is an operator <math>T\colon X\to X<math> of finite rank so that <math>\|Tx-x\|\leq\varepsilon<math>, for every <math>x\in K<math>.
Some other flavours of the AP are studied:
Let <math>X<math> be a Banach space and let <math>1\leq\lambda<\infty<math>. We say that <math>X<math> has the <math>\lambda<math>-approximation property (<math>\lambda<math>-AP), if, for every compact set <math>K\subset X<math> and every <math>\varepsilon>0<math>, there is an operator <math>T\colon X\to X<math> of finite rank so that <math>\|Tx-x\|\leq\varepsilon<math>, for every <math>x\in K<math>, and <math>\|T\|\leq\lambda<math>.
A Banach space space is said to have bounded approximation property (BAP), if it has the <math>\lambda<math>-AP for some <math>\lambda<math>.
A Banach space space is said to have metric approximation property (MAP), if it is 1-AP.
Examples
Every space with a Schauder basis has the AP (we can use the projections associated to the base as the <math>T<math>'s in the definition), thus a lot of spaces with the AP can be found. For example, the NaodW29-math2b13b76d68dbd50000000015 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).
- Lindenstrauss, J.; Tzafriri, L.: Classical Banach Spaces I, Sequence spaces, 1977.