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Proceedings of the American Mathematical Society

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

 

 

Stable unit balls in Orlicz spaces


Author: Antonio Suárez Granero
Journal: Proc. Amer. Math. Soc. 109 (1990), 97-104
MSC: Primary 46E30
DOI: https://doi.org/10.1090/S0002-9939-1990-1000154-6
MathSciNet review: 1000154
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Abstract: Let $ {L^\phi }(\mu )$ be an Orlicz space and $ X \subseteq {L^\phi }(\mu )$ an ideal such that $ {I_\phi }(f/\vert\vert f\vert\vert) = 1$ for each $ f \in X\backslash \left\{ 0 \right\}$. Then the unit ball $ {B_X}$ is stable, that is, the midpoint map $ {\Phi _{1/2}}:{B_X} \times {B_X} \to {B_X}$ defined by $ {\Phi _{1/2}}(x,y) = \tfrac{1}{2}(x + y)$, is open. In particular, $ {B_E}\phi $ is stable, $ {E^\phi }$ being the subspace of finite elements of $ {L^\phi }(\mu )$ (i.e., $ f \in {E^\phi }$ iff $ {I_\phi }(\lambda f) < + \infty $ for each $ \lambda > 0$), and $ {B_{{L^\phi }(\mu )}}$ is stable when $ \phi $ satisfies condition $ ({\Delta _2})$ or $ ({\delta _2})$, depending on the measure $ \mu $.


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DOI: https://doi.org/10.1090/S0002-9939-1990-1000154-6
Keywords: Stable sets, Orlicz spaces, unit ball
Article copyright: © Copyright 1990 American Mathematical Society