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

Vassiliki Farmaki

Studia Mathematica (1993)

  • Volume: 104, Issue: 1, page 61-74
  • ISSN: 0039-3223

Abstract

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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 ( 2 n ) ) equivalent to the usual basis of ℓ 1 (resp. as the elements with the sequence ( x ( 4 n - 2 ) - x ( 4 n ) ) 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.

How to cite

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Farmaki, Vassiliki. "Characterizations of elements of a double dual Banach space and their canonical reproductions." Studia Mathematica 104.1 (1993): 61-74. <http://eudml.org/doc/215959>.

@article{Farmaki1993,
abstract = {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.},
author = {Farmaki, Vassiliki},
journal = {Studia Mathematica},
keywords = {double dual of a separable Banach space; canonical reproductions; even- order duals; spreading model},
language = {eng},
number = {1},
pages = {61-74},
title = {Characterizations of elements of a double dual Banach space and their canonical reproductions},
url = {http://eudml.org/doc/215959},
volume = {104},
year = {1993},
}

TY - JOUR
AU - Farmaki, Vassiliki
TI - Characterizations of elements of a double dual Banach space and their canonical reproductions
JO - Studia Mathematica
PY - 1993
VL - 104
IS - 1
SP - 61
EP - 74
AB - 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.
LA - eng
KW - double dual of a separable Banach space; canonical reproductions; even- order duals; spreading model
UR - http://eudml.org/doc/215959
ER -

References

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  5. [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. Zbl0762.46006
  6. [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. [7] F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. (2) 30 (1929), 264-286. Zbl55.0032.04
  8. [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. [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. 

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