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From weak to strong types of $L_E^1$-convergence by the Bocce-criterion

Erik J. Balder·Maria Girardi·Vincent Jalby
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Abstract

Necessary and sufficient oscillation conditions are given for a weakly convergent sequence (resp. relatively weakly compact set) in the Bochner-Lebesgue space $\l1$ to be norm convergent (resp. relatively norm compact), thus extending the known results for $\rl1$. Similarly, necessary and sufficient oscillation conditions are given to pass from weak to limited (and also to Pettis-norm) convergence in $\l1$. It is shown that tightness is a necessary and sufficient condition to pass from limited to strong convergence. Other implications between several modes of convergence in $\l1$ are also studied.

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Erik J. Balder, Maria Girardi, Vincent Jalby. 1994-02-17. From weak to strong types of $L_E^1$-convergence by the Bocce-criterion. https://arxiv.org/abs/math/9402210

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