Characterizations of elements of a double dual Banach space and their canonical reproductions

• Autorzy: Vassiliki Farmaki
• Języki publikacji: EN

Abstrakty

EN

For every element x** in the double dual of a separable Banach space X there exists the sequence $(x^{(2 n)})$ of the canonical reproductions of x** in the even-order duals of X. In this paper we prove that every such sequence defines a spreading model for X. Using this result we characterize the elements of X**╲ X which belong to the class $B_1 (X)╲ B_{1/2}(X)$ (resp. to the class $B_{1/4}(X)$) as the elements with the sequence $(x^{(2n)})$ equivalent to the usual basis of $ℓ^1$ (resp. as the elements with the sequence $(x^{(4n-2)} - x^{(4n)})$ equivalent to the usual basis of $c_0$). Also, by analogous conditions but of isometric nature, we characterize the embeddability of $ℓ^1$ (resp. $c_0$) in X.

Kategorie tematyczne

• 46B25: Classical Banach spaces in the general theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1993
• Tom: 104
• Numer: 1
• Strony: 61-74

Daty

wydano

1993

otrzymano

1991-10-01

poprawiono

1992-08-05

Twórcy

autor

Vassiliki Farmaki

• Department of Mathematics, University of Athens, Panepistemiopolis-Ilisia, Gr-15784 Athens, Greece,

Bibliografia

• [1] B. Beauzamy et J. T. Lapreste, Modèles étalés des espaces de Banach, Travaux en Cours, Hermann, Paris 1984.
• [2] C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151-164.
• [3] A. Brunel and L. Sucheston, On J-convexity and ergodic superproperties of Banach spaces, Trans. Amer. Math. Soc. 204 (1975), 79-90.
• [4] V. Farmaki, $c_0$-subspaces and fourth dual types, Proc. Amer. Math. Soc. (2) 102 (1988), 321-328.
• [5] R. Haydon, E. Odell and H. Rosenthal, On certain classes of Baire-1 functions with applications to Banach space theory, in: Functional Analysis (Austin, Tex., 1987/1989), Lecture Notes in Math. 1470, Springer, 1991, 1-35.
• [6] B. Maurey, Types and $ℓ_1$-subspaces, in: Texas Functional Analysis Seminar 1982-1983, Longhorn Notes, Univ. Texas Press, Austin, Tex., 1983, 123-137.
• [7] F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. (2) 30 (1929), 264-286.
• [8] H. Rosenthal, Some remarks concerning unconditional basic sequences, in: Texas Functional Analysis Seminar 1982-1983, Longhorn Notes, Univ. Texas Press, Austin, Tex., 1983, 15-47.
• [9] H. Rosenthal, Double dual types and the Maurey characterization of Banach spaces containing $ℓ^1$, in: Texas Functional Analysis Seminar 1983-1984, Longhorn Notes, Univ. Texas Press, Austin, Tex., 1984, 1-37.