Approximation property

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]

In mathematics, 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, a lot of work in this area was done by Grothendieck (1955).

Later many other counterexamples were found. The space of bounded operators on \ell^2 does not have the approximation property (Szankowski). The spaces \ell^p for p\neq 2 and 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] If X is a Banach space this requirement becomes that 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.

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

References

  1. Megginson, Robert E. An Introduction to Banach Space Theory p. 336
  2. Schaefer p. 108
  3. Schaefer p. 110
  4. Schaefer p. 109
  5. Schaefer p. 115
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