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

Erik J. Balder; Maria Girardi; Vincent Jalby

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

Functional Analysis 2016-09-06 v1

Authors: Erik J. Balder , Maria Girardi , Vincent Jalby

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.

Keywords

Bochner-Lebesgue space weak convergence strong convergence Bocce criterion oscillation condition limited convergence

Cite

@article{arxiv.math/9402210,
  title  = {From weak to strong types of $L_E^1$-convergence by the Bocce-criterion},
  author = {Erik J. Balder and Maria Girardi and Vincent Jalby},
  journal= {arXiv preprint arXiv:math/9402210},
  year   = {2016}
}

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

Abstract

Keywords

Cite

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