Let be a Banach space and the algebra of bounded operators in . We say that has the approximation property if there is a sequence of finite rank (http://planetmath.org/RankLinearMapping) operators in such that
The fundamental fact about spaces with the approximation property is that every compact operator is the norm limit of finite rank operators.
Separable Hilbert spaces have the approximation property. Note however that compact operators on Hilbert spaces (not just separable ones) are always norm limit of finite rank operators.
The -spaces (http://planetmath.org/Lp) have the approximation property.
Theorem - If is a Banach space with a Schauder basis then it has the approximation property.
|Date of creation||2013-03-22 17:28:42|
|Last modified on||2013-03-22 17:28:42|
|Last modified by||asteroid (17536)|
|Synonym||approximation by finite rank operators|
|Defines||Schauder basis and approximation by finite rank operators|