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Weak and Norm Convergence of Sequences in Banach SpacesHymel, Arthur J. (Arthur Joseph) 12 1900 (has links)
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 ℓ¹ they coincide. We then show a theorem of Rosenthal's which
states that if {𝓍ₙ} is a bounded sequence in a Banach space, then {𝓍ₙ} has a subsequence
{𝓍'ₙ} satisfying one of the following two mutually exclusive alternatives; (i) {𝓍'ₙ} is weakly
Cauchy, or (ii) {𝓍'ₙ} is equivalent to the unit vector basis of ℓ¹.
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