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

Erik Balder; Maria Girardi; Vincent Jalby — 1994

Studia Mathematica

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....

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From weak to strong types of L E 1 -convergence by the Bocce criterion

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