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

Bell, Wayne C.

doi:10.1090/S0002-9939-1977-0482154-1

On the normability of the intersection of $L_{p}$ spaces
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Proc. Amer. Math. Soc. 66 (1977), 299-304 Request permission

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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