Mathematical concept
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. The space formula_1 of bounded operators on an infinite-dimensional Hilbert space formula_2 does not have the approximation property. The spaces formula_3 for formula_4 and formula_5 (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. 
For a locally convex space "X", the following are equivalent:
where formula_12 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 formula_16 and every formula_17, there is an operator formula_18 of finite rank so that formula_19, for every formula_20.
Related definitions.
Some other flavours of the AP are studied:
Let formula_21 be a Banach space and let formula_22. We say that "X" has the formula_23"-approximation property" (formula_23-AP), if, for every compact set formula_16 and every formula_17, there is an operator formula_27 of finite rank so that formula_28, for every formula_20, and formula_30.
A Banach space is said to have bounded approximation property (BAP), if it has the formula_23-AP for some formula_23.
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.
References.
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