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. 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, a lot of work in this area was done by Grothendieck (1955). Later many other counterexamples were found. The space of bounded operators on ℓ 2 {displaystyle ell ^{2}} does not have the approximation property (Szankowski). The spaces ℓ p {displaystyle ell ^{p}} for p ≠ 2 {displaystyle p eq 2} and c 0 {displaystyle c_{0}} (see Sequence space) have closed subspaces that do not have the approximation property. A locally convex topological vector space X {displaystyle 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. If X is a Banach space this requirement becomes that for every compact set K ⊂ X {displaystyle Ksubset X} and every ε > 0 {displaystyle varepsilon >0} , there is an operator T : X → X {displaystyle Tcolon X o X} of finite rank so that ‖ T x − x ‖ ≤ ε {displaystyle |Tx-x|leq varepsilon } , for every x ∈ K {displaystyle xin K} .