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Weak and Norm Convergence of Sequences in Banach Spaces

We study weak convergence of sequences in Banach spaces. In particular, we compare
the notions of weak and norm convergence. Although these modes of convergence usually
differ, we show that in โ„“ยน they coincide. We then show a theorem of Rosenthal's which
states that if {๐“โ‚™} is a bounded sequence in a Banach space, then {๐“โ‚™} has a subsequence
{๐“'โ‚™} satisfying one of the following two mutually exclusive alternatives; (i) {๐“'โ‚™} is weakly
Cauchy, or (ii) {๐“'โ‚™} is equivalent to the unit vector basis of โ„“ยน.

Identiferoai:union.ndltd.org:unt.edu/info:ark/67531/metadc500521
Date12 1900
CreatorsHymel, Arthur J. (Arthur Joseph)
ContributorsBator, Elizabeth M., Bilyeu, Russell Gene, Lewis, Paul Weldon
PublisherUniversity of North Texas
Source SetsUniversity of North Texas
LanguageEnglish
Detected LanguageEnglish
TypeThesis or Dissertation
Formatiii, 71 leaves, Text
RightsPublic, Hymel, Arthur J. (Arthur Joseph), Copyright, Copyright is held by the author, unless otherwise noted. All rights reserved.

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