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

Volume 111 / 1994

Erik J. Balder, , Studia Mathematica 111 (1994), 241-262 DOI: 10.4064/sm-111-3-241-262


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.


  • Erik J. Balder

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