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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

On the normability of the intersection of $ L\sb{p}$ spaces


Author: Wayne C. Bell
Journal: Proc. Amer. Math. Soc. 66 (1977), 299-304
MSC: Primary 46E99; Secondary 28A10
MathSciNet review: 0482154
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Abstract: The set $ {L_\omega } = \bigcap\nolimits_{p = 1}^\infty {{L_p}[0,1]} $ is not equal to $ {L_\infty }[0,1]$ since $ {L_\omega }$ contains the function $ - \ln x$. Using the theory of $ {L_p}$ spaces for finitely additive set functions developed by Leader [9] we will prove several necessary and sufficient conditions for the normability of a generalization of $ {L_\omega }$. These include the equality and finite dimensionality of all the $ {L_p}$ spaces, $ p \geqslant 1$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0482154-1
PII: S 0002-9939(1977)0482154-1
Keywords: Finitely additive set function, $ {L_p}$ spaces, normability
Article copyright: © Copyright 1977 American Mathematical Society