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- W2015442008 abstract "Let $E$ be a Banach space and, for any positive integer $n$, let ${cal P}(^nE)$ denote the Banach space of continuous $n$-homogeneous polynomials on $E$. Davie and Gamelin showed that the natural extension mapping from ${cal P}(^nE)$ to ${cal P}(^nE^{astast})$ is an isometry into the latter space. Here, we investigate when there is a natural isomorphism between ${cal P}(^nE)^{astast}$ and ${cal P}(^nE^{astast})$. Among other things, we show that if $E$ satisfies: break (a) no spreading model built on a normalised weakly null sequence has a lower $q$-estimate for any $q < infty,$ (b) $E^{ast}$ has RNP, and (c) $E^{ast}$ has the approximation property, then ${cal P}(^nE)$ has RNP for every $n$. Moreover, if $E$ satisfies (a) and is such that $E^{astast}$ has both the RNP and the approximation property, then ${cal P}(^nE)^{astast}$ and ${cal P}(^nE^{astast})$ are isomorphic for every $n$. We also exhibit a quasi-reflexive Banach space $E$ for which ${cal P}(^nE)^{astast}$ and ${cal P}(^nE^{astast})$ are isomorphic for every $n$. Related work has been done recently by (i) M. Gonzalez, (ii) M. Valdivia, and (iii) J. Jaramillo, A. Prieto, and I. Zalduendo." @default.
- W2015442008 created "2016-06-24" @default.
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- W2015442008 date "1997-12-01" @default.
- W2015442008 modified "2023-09-28" @default.
- W2015442008 title "$Q$-Reflexive Banach Spaces" @default.
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- W2015442008 doi "https://doi.org/10.1216/rmjm/1181071856" @default.
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