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

Balder, Erik; Girardi, Maria; Jalby, Vincent

From weak to strong types of

L_{E}^{1}

-convergence by the Bocce criterion

Balder, Erik ; Girardi, Maria ; Jalby, Vincent

Studia Mathematica, Tome 108 (1994), p. 241-262 / Harvested from The Polish Digital Mathematics Library

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Résumé

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.

Publié le : 1994-01-01

Zbl 0809.28006

EUDML-ID : urn:eudml:doc:216131

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     author = {Erik Balder and Maria Girardi and Vincent Jalby},
     title = {From weak to strong types of $L^{1}\_{E}$-convergence by the Bocce criterion},
     journal = {Studia Mathematica},
     volume = {108},
     year = {1994},
     pages = {241-262},
     zbl = {0809.28006},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv111i3p241bwm}
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Balder, Erik; Girardi, Maria; Jalby, Vincent. From weak to strong types of $L^{1}_{E}$-convergence by the Bocce criterion. Studia Mathematica, Tome 108 (1994) pp. 241-262. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv111i3p241bwm/

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