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

Erik Balder; Maria Girardi; Vincent Jalby

Studia Mathematica (1994)

  • Volume: 111, Issue: 3, page 241-262
  • ISSN: 0039-3223

Abstract

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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. Other implications between several modes of convergence in E 1 are also studied.

How to cite

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Balder, Erik, Girardi, Maria, and Jalby, Vincent. "From weak to strong types of $L^{1}_{E}$-convergence by the Bocce criterion." Studia Mathematica 111.3 (1994): 241-262. <http://eudml.org/doc/216131>.

@article{Balder1994,
abstract = {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.},
author = {Balder, Erik, Girardi, Maria, Jalby, Vincent},
journal = {Studia Mathematica},
keywords = {weak convergence; Radon-Nikodým property; Bocce criterion; oscillation condition; limited convergence; strong convergence; tightness; Pettis norm},
language = {eng},
number = {3},
pages = {241-262},
title = {From weak to strong types of $L^\{1\}_\{E\}$-convergence by the Bocce criterion},
url = {http://eudml.org/doc/216131},
volume = {111},
year = {1994},
}

TY - JOUR
AU - Balder, Erik
AU - Girardi, Maria
AU - Jalby, Vincent
TI - From weak to strong types of $L^{1}_{E}$-convergence by the Bocce criterion
JO - Studia Mathematica
PY - 1994
VL - 111
IS - 3
SP - 241
EP - 262
AB - 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.
LA - eng
KW - weak convergence; Radon-Nikodým property; Bocce criterion; oscillation condition; limited convergence; strong convergence; tightness; Pettis norm
UR - http://eudml.org/doc/216131
ER -

References

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