Extremal Tests for Weak Convergence of Sequences and Series
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 ∑n∣x*x n∣ < ∞ for each extreme point x* of B x* is unconditionally convergent.
KeywordsAssure Hull Harness
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