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Zbl 0722.46014
Suárez Granero, Antonio
Stable unit balls in Orlicz spaces. (English)
[J] Proc. Am. Math. Soc. 109, No.1, 97-104 (1990). ISSN 0002-9939; ISSN 1088-6826/e

The author proves that if $L\sp{\phi}(\mu)$ is an Orlicz space and $X\subset L\sp{\phi}(\mu)$ an ideal such that for each $f\in X\setminus \{0\}$ the modular $I\sb{\phi}(f/\Vert f\Vert)=1$, then the closed unit ball $B\sb X$ of X is stable, that is, the midpoint map $(x,y)\to (x+y)$ from $B\sb X\times B\sb X$ into $B\sb X$ is open. Here the $\Delta$ ${}\sb 2$ condition is not assumed as it is in the classical case. Stable sets have been studied by {\it A. Clausing} and {\it S. Papadopoulou} [Math. Ann. 231, 193-203 (1978; Zbl 0349.46002)], {\it R. Grzaslewicz} [Bull. Polish. Acad. Sci. Math. 33, 277-283 (1985; Zbl 0597.46024)], and {\it S. Papadopoulou} [Math. Ann. 229, 193-200 (1977; Zbl 0339.46001)].
[P.-Y.Lee (Singapore)]

MSC 2000:: *46E30 Spaces of measurable functions
46B20 Geometry and structure of normed spaces
47L07 Convex sets and cones of operators

Keywords: Orlicz space; modular; midpoint map; $\Delta \sb 2$ condition is not assumed; Stable sets

Citations: Zbl 0349.46002; Zbl 0597.46024; Zbl 0339.46001

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