Weak and Norm Convergence of Sequences in Banach Spaces

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Description

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.

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iii, 71 leaves

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Hymel, Arthur J. (Arthur Joseph) December 1993.

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This thesis is part of the collection entitled: UNT Theses and Dissertations and was provided by UNT Libraries to Digital Library, a digital repository hosted by the UNT Libraries. It has been viewed 21 times . More information about this thesis can be viewed below.

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  • Hymel, Arthur J. (Arthur Joseph)

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Description

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.

Physical Description

iii, 71 leaves

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UNT Theses and Dissertations

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  • December 1993

Added to The UNT Digital Library

  • March 9, 2015, 8:15 a.m.

Description Last Updated

  • June 28, 2017, 1:28 p.m.

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Citations, Rights, Re-Use

Hymel, Arthur J. (Arthur Joseph). Weak and Norm Convergence of Sequences in Banach Spaces, thesis, December 1993; Denton, Texas. (digital.library.unt.edu/ark:/67531/metadc500521/: accessed October 24, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; .