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).

Later many other counterexamples were found. Space of bounded operators on <math>l_2</math> does not have the approximation property (Shankovskii). The spaces <math>l_p</math> for <math>p\not =2</math> and <math>c_0</math> (see Sequence space) have closed subspaces, which don't have the approximation property.

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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.

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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 <math>l^p</math> spaces, or the symmetric Tsirelson space.

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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.
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