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.
KeywordsBanach Space Extreme Point Weak Convergence Separable Banach Space Compact Convex Subset
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- Alfsen, E. M. 1971. Compact Convex Sets and Boundary Integrals. Ergebnisse der Mathematik und ihrer Grenzgebiete, Volume 57. Berlin: Springer-Verlag.Google Scholar
- Bourgin, R. 1983. Geometric Aspects of Convex Sets with the Radon-Nikodym Property, Volume 993, Springer Lecture Notes in Mathematics. Berlin: Springer-Verlag.Google Scholar
- Choquet, G. 1969. Lectures on Analysis, Lecture Notes in Mathematics. New York: W. A. Benjamin.Google Scholar
- Phelps, R. R. 1966. Lectures on Choquet’s theorem. Van Nostrand Math. Studies No. 7. Princeton: Van Nostrand.Google Scholar