A characterization of Banach spaces containing

Author:
Haskell Rosenthal

Journal:
J. Amer. Math. Soc. **7** (1994), 707-748

MSC:
Primary 46B99; Secondary 46B15, 46B25

MathSciNet review:
1242455

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Abstract: A subsequence principle is obtained, characterizing Banach spaces containing , in the spirit of the author's 1974 characterization of Banach spaces containing .

**Definition**. A sequence in a Banach space is called *strongly summing* (s.s.) if is a weak-Cauchy basic sequence so that whenever scalars satisfy , then converges.

A simple permanence property: if is an (s.s.) basis for a Banach space and are its biorthogonal functionals in , then is a non-trivial weak-Cauchy sequence in ; hence fails to be weakly sequentially complete. (A weak-Cauchy sequence is called *non-trivial* if it is *non-weakly convergent*.)

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

*Remark*. The two alternatives of the theorem are easily seen to be mutually exclusive.

**Corollary 1**. *A Banach space* *contains no isomorph of* *if and only if every non-trivial weak-Cauchy sequence in* *has an* (s.s.) *subsequence*.

Combining the - and -Theorems, we obtain

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

The proof of the theorem involves a careful study of differences of bounded semi-continuous functions. The results of this study may be of independent interest.

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Additional Information

DOI:
https://doi.org/10.1090/S0894-0347-1994-1242455-4

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

Article copyright:
© Copyright 1994
American Mathematical Society