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A subsequence principle characterizing Banach spaces containing $ c_0$

Author: Haskell Rosenthal
Journal: Bull. Amer. Math. Soc. 30 (1994), 227-233
MSC (2000): Primary 46B15
MathSciNet review: 1249355
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Abstract: The notion of a strongly summing sequence is introduced. Such a sequence is weak-Cauchy, a basis for its closed linear span, and has the crucial property that the dual of this span is not weakly sequentially complete. The main result is:

Theorem. Every non-trivial weak-Cauchy sequence in a (real or complex) Banach space has either a strongly summing sequence or a convex block basis equivalent to the summing basis.

(A weak-Cauchy sequence is called non-trivial if it is non-weakly convergent.) The following characterization of spaces containing $ {c_0}$ is thus obtained, in the spirit of the author's 1974 subsequence principle characterizing Banach spaces containing $ {\ell ^1}$.

Corollary 1. A Banach space B contains no isomorph of $ {c_0}$ if and only if every non-trivial weak-Cauchy sequence in B has a strongly summing subsequence.

Combining the $ {c_0}$-and $ {\ell ^1}$-theorems, one obtains

Corollary 2. If B is a non-reflexive Banach space such that $ {X^{\ast}}$ is weakly sequentially complete for all linear subspaces X of B, then $ {c_0}$ embeds in B.

References [Enhancements On Off] (What's this?)

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Keywords: Weakly sequentially complete dual, convex block basis, the $ {\ell ^1}$-theorem, differences of semi-continuous functions
Article copyright: © Copyright 1994 American Mathematical Society

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