approximation property

Let Y be a Banach spaceMathworldPlanetmath and B(Y) the algebra of bounded operatorsMathworldPlanetmathPlanetmath in Y. We say that Y has the approximation propertyMathworldPlanetmath if there is a sequencePlanetmathPlanetmath (Pn) of finite rank ( operatorsMathworldPlanetmath in B(Y) such that


The fundamental fact about spaces with the approximation property is that every compact operatorMathworldPlanetmath is the norm limit of finite rank operators.

Theorem - Let X be a normed vector spacePlanetmathPlanetmath and Y a Banach space with the approximation property. Then every compact operator T:XY 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 separablePlanetmathPlanetmath ones) are always norm limit of finite rank operators.

  • The p-spaces ( have the approximation property.


Theorem - If Y 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