A characterization of Banach spaces containing

Author:
Haskell Rosenthal

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

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

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

MathSciNet review:
1242455

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Abstract | References | Similar Articles | Additional Information

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.

**[Be]**Steven F. Bellenot,*More quasireflexive subspaces*, Proc. Amer. Math. Soc.**101**(1987), no. 4, 693–696. MR**911035**, https://doi.org/10.1090/S0002-9939-1987-0911035-8**[Bes-P]**C. Bessaga and A. Pełczyński,*On bases and unconditional convergence of series in Banach spaces*, Studia Math.**17**(1958), 151–164. MR**0115069****[Bo-De]**J. Bourgain and F. Delbaen,*A class of special \cal𝐿_{∞} spaces*, Acta Math.**145**(1980), no. 3-4, 155–176. MR**590288**, https://doi.org/10.1007/BF02414188**[Bo-R]**J. Bourgain and H. P. Rosenthal,*Geometrical implications of certain finite-dimensional decompositions*, Bull. Soc. Math. Belg. Sér. B**32**(1980), no. 1, 57–82. MR**682992****[Do]**Leonard E. Dor,*On sequences spanning a complex 𝑙₁ space*, Proc. Amer. Math. Soc.**47**(1975), 515–516. MR**0358308**, https://doi.org/10.1090/S0002-9939-1975-0358308-X**[E]**J. Elton,*Weakly null normalized sequences in Banach spaces*, Doctoral Thesis, Yale University, 1978.**[F]**Catherine Finet,*Subspaces of Asplund Banach spaces with the point continuity property*, Israel J. Math.**60**(1987), no. 2, 191–198. MR**931876**, https://doi.org/10.1007/BF02790791**[Go]**W.T. Gowers,*A space not containing**or a reflexive subspace*, preprint.**[HOR]**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**, https://doi.org/10.1007/BFb0090209**[JR]**W. B. Johnson and H. P. Rosenthal,*On 𝜔*-basic sequences and their applications to the study of Banach spaces*, Studia Math.**43**(1972), 77–92. MR**0310598****[KL]**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**, https://doi.org/10.1090/S0002-9947-1990-0946424-3**[OR]**E. Odell and H. P. Rosenthal,*A double-dual characterization of separable Banach spaces containing 𝑙¹*, Israel J. Math.**20**(1975), no. 3-4, 375–384. MR**0377482**, https://doi.org/10.1007/BF02760341**[P1]**A. Pełczyński,*A connection between weakly unconditional convergence and weakly completeness of Banach spaces*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys.**6**(1958), 251–253 (unbound insert) (English, with Russian summary). MR**0115072****[P2]**A. Pełczyński,*Banach spaces on which every unconditionally converging operator is weakly compact*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.**10**(1962), 641–648. MR**0149295****[R1]**Haskell P. Rosenthal,*A characterization of Banach spaces containing 𝑙¹*, Proc. Nat. Acad. Sci. U.S.A.**71**(1974), 2411–2413. MR**0358307****[R2]**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**, https://doi.org/10.1090/S0002-9904-1978-14521-2**[R3]**Haskell Rosenthal,*Weak*-Polish Banach spaces*, J. Funct. Anal.**76**(1988), no. 2, 267–316. MR**924462**, https://doi.org/10.1016/0022-1236(88)90039-0**[R4]**H. P. Rosenthal,*Some aspects of the subspace structure of infinite-dimensional Banach spaces*, Approximation theory and functional analysis (College Station, TX, 1990) Academic Press, Boston, MA, 1991, pp. 151–176. MR**1090555****[R5]**-,*Differences of bounded semi-continuous functions*(in preparation).**[R6]**-,*Boundedly complete weak-Cauchy basic sequences in Banach spaces with the PCP*(to appear).

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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