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

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.