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Journal Article

# Stable Unit Balls in Orlicz Spaces

Antonio Suárez Granero
Proceedings of the American Mathematical Society
Vol. 109, No. 1 (May, 1990), pp. 97-104
DOI: 10.2307/2048367
Stable URL: http://www.jstor.org/stable/2048367
Page Count: 8

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Topics: Orlicz space, Unit ball, Banach space, Mathematical functions, Mathematics

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## Abstract

Let Lφ(μ) be an Orlicz space and $X \subseteq L^\phi(\mu)$ an ideal such that Iφ(f/|f|) = 1 for each $f \in X\backslash\{0 \}$. Then the unit ball BX is stable, that is, the midpoint map Φ1/2: BX × BX → BX defined by Φ1/2(x, y) = 1/2(x + y), is open. In particular, BEφ is stable, Eφ being the subspace of finite elements of Lφ(μ) (i.e., $f \in E^\phi \operatorname{iff} I_\phi(\lambda f) < +\infty$ for each $\lambda > 0$), and BLφ(μ) is stable when φ satisfies condition (Δ2) or (δ2), depending on the measure μ.

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