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

Studia Mathematica (1993)

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

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## Abstract

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For every element x** in the double dual of a separable Banach space X there exists the sequence $\left({x}^{\left(2n\right)}\right)$ 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 ${B}_{1}\left(X\right)╲{B}_{1/2}\left(X\right)$ (resp. to the class ${B}_{1/4}\left(X\right)$) as the elements with the sequence $\left({x}^{\left(2n\right)}\right)$ equivalent to the usual basis of ${\ell }^{1}$ (resp. as the elements with the sequence $\left({x}^{\left(4n-2\right)}-{x}^{\left(4n\right)}\right)$ equivalent to the usual basis of ${c}_{0}$). Also, by analogous conditions but of isometric nature, we characterize the embeddability of ${\ell }^{1}$ (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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6. [6] B. Maurey, Types and ${\ell }_{1}$-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 ${\ell }^{1}$, in: Texas Functional Analysis Seminar 1983-1984, Longhorn Notes, Univ. Texas Press, Austin, Tex., 1984, 1-37.

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