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THE ${\mathcal{K}}_{\mathit{up}}$-APPROXIMATION PROPERTY AND ITS DUALITY
Published online by Cambridge University Press: 20 November 2014
Abstract
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.
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- Research Article
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- © 2014 Australian Mathematical Publishing Association Inc.
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