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

Erik J. Balder; Maria Girardi; Vincent Jalby

ArticleOriginal scientific text

Title

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

Authors ¹, ², ³

Affiliations

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

Abstract

Necessary and sufficient oscillation conditions are given for a weakly convergent sequence (resp. relatively weakly compact set) in the Bochner-Lebesgue space

ℒ_{E}^{1}

to be norm convergent (resp. relatively norm compact), thus extending the known results for

ℒ_{ℝ}^{1}

. Similarly, necessary and sufficient oscillation conditions are given to pass from weak to limited (and also to Pettis-norm) convergence in

ℒ_{E}^{1}

. 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_{E}^{1}

are also studied.

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Title

From weak to strong types of LE1-convergence by the Bocce criterion

Affiliations

Abstract

Bibliography

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