Approximation property

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.

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.

of bounded operators on an infinite-dimensional Hilbert space

does not have the approximation property.

(see Sequence space) have closed subspaces that do not have the approximation property.

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.

[3] For a locally convex space X, the following are equivalent:[3] 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

Some other flavours of the AP are studied: Let

-AP), if, for every compact set

A Banach space is said to have bounded approximation property (BAP), if it has the

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.