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THE
${\mathcal{K}}_{\mathit{up}}$-APPROXIMATION PROPERTY AND ITS DUALITY
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 98 / Issue 3 / June 2015
- Published online by Cambridge University Press:
- 20 November 2014, pp. 364-374
- Print publication:
- June 2015
-
- Article
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We introduce an approximation property (
${\mathcal{K}}_{\mathit{up}}$-AP, 
$1\leq p<\infty$), which is weaker than the classical approximation property, and discover the duality relationship between the 
${\mathcal{K}}_{\mathit{up}}$-AP and the 
${\mathcal{K}}_{p}$-AP. More precisely, we prove that for every 
$1<p<\infty$, if the dual space 
$X^{\ast }$ of a Banach space 
$X$ has the 
${\mathcal{K}}_{\mathit{up}}$-AP, then 
$X$ has the 
${\mathcal{K}}_{p}$-AP, and if 
$X^{\ast }$ has the 
${\mathcal{K}}_{p}$-AP, then 
$X$ has the 
${\mathcal{K}}_{\mathit{up}}$-AP. As a consequence, it follows that every Banach space has the 
${\mathcal{K}}_{u2}$-AP and that for every 
$1<p<\infty$, 
$p\neq 2$, there exists a separable reflexive Banach space failing to have the 
${\mathcal{K}}_{\mathit{up}}$-AP.
