approximation property
Let $Y$ be a Banach space^{} and $B(Y)$ the algebra of bounded operators^{} in $Y$. We say that $Y$ has the approximation property^{} if there is a sequence^{} $({P}_{n})$ of finite rank (http://planetmath.org/RankLinearMapping) operators^{} in $B(Y)$ such that
$${P}_{n}y\u27f6y{\forall}_{y\in Y}$$ 
i.e. $({P}_{n})$ 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 $X$ be a normed vector space^{} and $Y$ a Banach space with the approximation property. Then every compact operator $T:X\u27f6Y$ is the norm limit of operators of finite rank.
Examples :

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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.

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The ${\mathrm{\ell}}^{p}$spaces (http://planetmath.org/Lp) have the approximation property.
Moreover,
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  20130322 17:28:42 
Last modified on  20130322 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 