A subsequence principle characterizing Banach spaces containing

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 is thus obtained, in the spirit of the author's 1974 subsequence principle characterizing Banach spaces containing .

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

Combining the -and -theorems, one obtains

**Corollary 2**. *If B is a non-reflexive Banach space such that is weakly sequentially complete for all linear subspaces X of B, then embeds in B*.

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Additional Information

DOI:
https://doi.org/10.1090/S0273-0979-1994-00494-X

Keywords:
Weakly sequentially complete dual,
convex block basis,
the -theorem,
differences of semi-continuous functions

Article copyright:
© Copyright 1994
American Mathematical Society