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

Erik BalderMaria GirardiVincent 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 E 1 to be norm convergent (resp. relatively norm compact), thus extending the known results for 1 . Similarly, necessary and sufficient oscillation conditions are given to pass from weak to limited (and also to Pettis-norm) convergence in E 1 . It is shown that tightness is a necessary and sufficient condition to pass from limited to strong convergence....

On Weak Compactness and Lower Closure Results for Pettis Integrable (Multi)Functions

Erik J. BalderAnna Rita Sambucini — 2004

Bulletin of the Polish Academy of Sciences. Mathematics

In [4, 5, 7] an abstract, versatile approach was given to sequential weak compactness and lower closure results for scalarly integrable functions and multifunctions. Its main tool is an abstract version of the Komlós theorem, which applies to scalarly integrable functions. Here it is shown that this same approach also applies to Pettis integrable multifunctions, because the abstract Komlós theorem can easily be extended so as to apply to generalized Pettis integrable functions. Some results in the...

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