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. The space of bounded operators on formula_1 does not have the approximation property (Szankowski). The spaces formula_2 for formula_3 and formula_4 (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_11 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_15 and every formula_16, there is an operator  formula_17 of finite rank so that formula_18, for every formula_19. Related definitions. Some other flavours of the AP are studied: Let formula_20 be a Banach space and let formula_21. We say that "X" has the formula_22"-approximation property" (formula_22-AP), if, for every compact set formula_15 and every formula_16, there is an operator formula_26 of finite rank so that formula_27, for every formula_19, and formula_29. A Banach space is said to have bounded approximation property (BAP), if it has the formula_22-AP for some formula_22. 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.