Editing Approximation property (section)

== Related definitions ==
Some other flavours of the AP are studied:

Let <math>X</math> be a Banach space and let <math>1\leq\lambda<\infty</math>. We say that ''X'' has the <math>\lambda</math>''-approximation property'' (<math>\lambda</math>'''-AP'''), if, 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>, and <math>\|T\|\leq\lambda</math>.

A Banach space is said to have '''bounded approximation property''' ('''BAP'''), if it has the <math>\lambda</math>-AP for some <math>\lambda</math>.

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.