approximation property
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
i.e. converges in the strong operator topology to the identity operator
.
The fundamental fact about spaces with the approximation property is that every compact operator is the norm limit of finite rank operators.
Theorem - Let be a normed vector space and a Banach space with the approximation property. Then every compact operator is the norm limit of operators of finite rank.
Examples :
-
•
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.
Moreover,
Theorem - If is a Banach space with a Schauder basis then it has the approximation property.
Title | approximation property |
---|---|
Canonical name | ApproximationProperty |
Date of creation | 2013-03-22 17:28:42 |
Last modified on | 2013-03-22 17:28:42 |
Owner | asteroid (17536) |
Last modified by | asteroid (17536) |
Numerical id | 5 |
Author | asteroid (17536) |
Entry type | Definition |
Classification | msc 46B99 |
Synonym | approximation by finite rank operators |
Defines | Schauder basis and approximation by finite rank operators |