Extremal Tests for Weak Convergence of Sequences and Series

  • Joseph Diestel
Part of the Graduate Texts in Mathematics book series (GTM, volume 92)


This chapter has two theorems as foci. The first, due to the enigmatic Rainwater, states that for a bounded sequence (x n) in a Banach space X to converge weakly to the point x, it is necessary and sufficient that x*x = lim n x*x n hold for each extreme point x* of B x* . The second improves the Bessaga-Pelczynski criterion for detecting c 0’s absence; thanks to Elton, we are able to prove that in a Banach space X without a copy of c 0 inside it, any series ∑ n x n for which ∑nx*x n∣ < ∞ for each extreme point x* of B x* is unconditionally convergent.


Banach Space Extreme Point Weak Convergence Separable Banach Space Compact Convex Subset 
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Copyright information

© Springer-Verlag New York, Inc. 1984

Authors and Affiliations

  • Joseph Diestel
    • 1
  1. 1.Department of Math SciencesKent State UniversityKentUSA

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