A characterization of Banach spaces containing
Author:
Haskell Rosenthal
Journal:
J. Amer. Math. Soc. 7 (1994), 707748
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 weakCauchy 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 nontrivial weakCauchy sequence in ; hence fails to be weakly sequentially complete. (A weakCauchy sequence is called nontrivial if it is nonweakly convergent.) Theorem. Every nontrivial weakCauchy 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 nontrivial weakCauchy sequence in has an (s.s.) subsequence. Combining the  and Theorems, we obtain Corollary 2. If is a nonreflexive 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 semicontinuous functions. The results of this study may be of independent interest.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S08940347199412424554
PII:
S 08940347(1994)12424554
Keywords:
Weakly sequentially complete dual,
convex block basis,
the Theorem,
differences of semicontinuous functions
Article copyright:
© Copyright 1994
American Mathematical Society
