A subsequence principle characterizing Banach spaces containing $c_0$
HTML articles powered by AMS MathViewer
- by Haskell Rosenthal PDF
- Bull. Amer. Math. Soc. 30 (1994), 227-233 Request permission
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
- Steven F. Bellenot, More quasireflexive subspaces, Proc. Amer. Math. Soc. 101 (1987), no. 4, 693–696. MR 911035, DOI 10.1090/S0002-9939-1987-0911035-8
- C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151–164. MR 115069, DOI 10.4064/sm-17-2-151-164
- J. Bourgain and F. Delbaen, A class of special ${\cal L}_{\infty }$ spaces, Acta Math. 145 (1980), no. 3-4, 155–176. MR 590288, DOI 10.1007/BF02414188
- Leonard E. Dor, On sequences spanning a complex $l_{1}$ space, Proc. Amer. Math. Soc. 47 (1975), 515–516. MR 358308, DOI 10.1090/S0002-9939-1975-0358308-X
- R. Haydon, E. Odell, and H. Rosenthal, On certain classes of Baire-$1$ functions with applications to Banach space theory, Functional analysis (Austin, TX, 1987/1989) Lecture Notes in Math., vol. 1470, Springer, Berlin, 1991, pp. 1–35. MR 1126734, DOI 10.1007/BFb0090209
- A. S. Kechris and A. Louveau, A classification of Baire class $1$ functions, Trans. Amer. Math. Soc. 318 (1990), no. 1, 209–236. MR 946424, DOI 10.1090/S0002-9947-1990-0946424-3
- Haskell P. Rosenthal, A characterization of Banach spaces containing $l^{1}$, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411–2413. MR 358307, DOI 10.1073/pnas.71.6.2411
- Haskell P. Rosenthal, Some recent discoveries in the isomorphic theory of Banach spaces, Bull. Amer. Math. Soc. 84 (1978), no. 5, 803–831. MR 499730, DOI 10.1090/S0002-9904-1978-14521-2 —, A characterization of Banach spaces containing ${c_0}$, J. Amer. Math. Soc. (to appear). —, Differences of bounded semi-continuous functions, in preparation.
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 30 (1994), 227-233
- MSC (2000): Primary 46B15
- DOI: https://doi.org/10.1090/S0273-0979-1994-00494-X
- MathSciNet review: 1249355