Editing Approximation property (section)

== Definition ==
A [[locally convex]] topological vector space ''X'' is said to have '''the approximation property''', if the identity map can be approximated, uniformly on [[Relatively compact subspace|precompact set]]s, by continuous linear maps of finite rank.{{sfn | Schaefer|Wolff| 1999 | p=108-115}} 

For a locally convex space ''X'', the following are equivalent:{{sfn | Schaefer|Wolff| 1999 | p=108-115}}
# ''X'' has the approximation property;
# the closure of <math>X^{\prime} \otimes X</math> in <math>\operatorname{L}_p(X, X)</math> contains the identity map <math>\operatorname{Id} : X \to X</math>;
# <math>X^{\prime} \otimes X</math> is dense in <math>\operatorname{L}_p(X, X)</math>;
# for every locally convex space ''Y'', <math>X^{\prime} \otimes Y</math> is dense in <math>\operatorname{L}_p(X, Y)</math>;
# for every locally convex space ''Y'', <math>Y^{\prime} \otimes X</math> is dense in <math>\operatorname{L}_p(Y, X)</math>;
where <math>\operatorname{L}_p(X, Y)</math> 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]] <math>K\subset X</math> and every <math>\varepsilon>0</math>, there is an [[operator (mathematics)|operator]] <math>T\colon X\to X</math> of finite rank so that <math>\|Tx-x\|\leq\varepsilon</math>, for every <math>x \in K</math>.