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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the second dual of the Lorentz space
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by Pratibha G. Ghatage and Brian M. Scott PDF
Proc. Amer. Math. Soc. 92 (1984), 239-244 Request permission

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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Additional Information
  • © Copyright 1984 American Mathematical Society
  • 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