Approximation property

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The construction of a Banach space without the approximation property earned Per Enflo a live goose in 1972, which had been promised by Stanislaw Mazur (left) in 1936. MazurGes.jpg
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.

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.

Contents

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 of bounded operators on does not have the approximation property (Szankowski). The spaces for and (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]

For a locally convex space X, the following are equivalent: [2]

  1. X has the approximation property;
  2. the closure of in contains the identity map ;
  3. is dense in ;
  4. for every locally convex space Y, is dense in ;
  5. for every locally convex space Y, is dense in ;

where 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 and every , there is an operator of finite rank so that , for every .

Some other flavours of the AP are studied:

Let be a Banach space and let . We say that X has the -approximation property (-AP), if, for every compact set and every , there is an operator of finite rank so that , for every , and .

A Banach space is said to have bounded approximation property (BAP), if it has the -AP for some .

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

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References

  1. Megginson, Robert E. An Introduction to Banach Space Theory p. 336
  2. 1 2 3 4 5 Schaefer & Wolff 1999, p. 108-115.

Bibliography