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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On the second dual of the Lorentz space


Authors: Pratibha G. Ghatage and Brian M. Scott
Journal: Proc. Amer. Math. Soc. 92 (1984), 239-244
MSC: Primary 46E30
DOI: https://doi.org/10.1090/S0002-9939-1984-0754711-X
MathSciNet review: 754711
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Abstract: If $ \phi (t) = {t^{1/p}}(p > 1)$ and $ (X,\mathcal{S},\mu )$ is a completely nonatomic finite measure space, then the dual of the Lorentz space $ {N_\phi }$ is denoted by $ {M_\phi }$ and the closure of the simple functions in $ {M_\phi }$ by $ M_\phi ^0$. It is known that $ {(M_\phi ^0)^ * } = {N_\phi }$. In this note we show that given a positive number $ \beta < 1$ it is possible to construct a set of contractive embeddings of $ ({l_\infty }/{c_0})$ into $ {({M_\phi }/M_\phi ^0)^ * }$, each of which is bounded below by $ M = M(\beta ) \to 1\;{\text{as}}\;\beta \to {{\text{0}}^ + }$. The union of the ranges of these embeddings is a total set in $ {({M_\phi }/M_\phi ^0)^ * }$.


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DOI: https://doi.org/10.1090/S0002-9939-1984-0754711-X
Keywords: Lorentz space
Article copyright: © Copyright 1984 American Mathematical Society