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Proceedings of the American Mathematical Society

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

 
 

 

On the normability of the intersection of $L_{p}$ spaces


Author: Wayne C. Bell
Journal: Proc. Amer. Math. Soc. 66 (1977), 299-304
MSC: Primary 46E99; Secondary 28A10
DOI: https://doi.org/10.1090/S0002-9939-1977-0482154-1
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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Keywords: Finitely additive set function, <IMG WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${L_p}$"> spaces, normability
Article copyright: © Copyright 1977 American Mathematical Society