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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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Keywords: Lorentz space
Article copyright: © Copyright 1984 American Mathematical Society