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From weak to strong types of $L^{1}_{E}$-convergence by the Bocce criterion

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Necessary and sufficient oscillation conditions are given for a weakly convergent sequence (resp. relatively weakly compact set) in the Bochner-Lebesgue space $ℒ^{1}_{E}$ to be norm convergent (resp. relatively norm compact), thus extending the known results for $ℒ^{1}_{ℝ}$. Similarly, necessary and sufficient oscillation conditions are given to pass from weak to limited (and also to Pettis-norm) convergence in $ℒ^{1}_{E}$. 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 $ℒ^{1}_{E}$ are also studied.
Słowa kluczowe
  • Mathematical Institute, University of Utrecht, P.O. Box 80.010, 3508 Ta Utrecht, The Netherlands
  • Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208, U.S.A.
  • Département de Mathématiques, Université Montpellier II, 34095 Montpellier Cedex 05, France
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