A subsequence characterization of sequences spanning isomorphically polyhedral Banach spaces

G. Androulakis

Studia Mathematica (1998)

  • Volume: 127, Issue: 1, page 65-80
  • ISSN: 0039-3223

Abstract

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Let (x_n) be a sequence in a Banach space X which does not converge in norm, and let E be an isomorphically precisely norming set for X such that (*) ∑_n |x*(x_{n+1} - x_n)| < ∞, ∀x* ∈ E. Then there exists a subsequence of (x_n) which spans an isomorphically polyhedral Banach space. It follows immediately from results of V. Fonf that the converse is also true: If Y is a separable isomorphically polyhedral Banach space then there exists a normalized M-basis (x_n) which spans Y and there exists an isomorphically precisely norming set E for Y such that (*) is satisfied. As an application of this subsequence characterization of sequences spanning isomorphically polyhedral Banach spaces we obtain a strengthening of a result of J. Elton, and an Orlicz-Pettis type result.

How to cite

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Androulakis, G.. "A subsequence characterization of sequences spanning isomorphically polyhedral Banach spaces." Studia Mathematica 127.1 (1998): 65-80. <http://eudml.org/doc/216460>.

@article{Androulakis1998,
abstract = {Let (x\_n) be a sequence in a Banach space X which does not converge in norm, and let E be an isomorphically precisely norming set for X such that (*) ∑\_n |x*(x\_\{n+1\} - x\_n)| < ∞, ∀x* ∈ E. Then there exists a subsequence of (x\_n) which spans an isomorphically polyhedral Banach space. It follows immediately from results of V. Fonf that the converse is also true: If Y is a separable isomorphically polyhedral Banach space then there exists a normalized M-basis (x\_n) which spans Y and there exists an isomorphically precisely norming set E for Y such that (*) is satisfied. As an application of this subsequence characterization of sequences spanning isomorphically polyhedral Banach spaces we obtain a strengthening of a result of J. Elton, and an Orlicz-Pettis type result.},
author = {Androulakis, G.},
journal = {Studia Mathematica},
keywords = {isomorphically polyhedral Banach space; normalized M-basis; Orlicz-Pettis type result},
language = {eng},
number = {1},
pages = {65-80},
title = {A subsequence characterization of sequences spanning isomorphically polyhedral Banach spaces},
url = {http://eudml.org/doc/216460},
volume = {127},
year = {1998},
}

TY - JOUR
AU - Androulakis, G.
TI - A subsequence characterization of sequences spanning isomorphically polyhedral Banach spaces
JO - Studia Mathematica
PY - 1998
VL - 127
IS - 1
SP - 65
EP - 80
AB - Let (x_n) be a sequence in a Banach space X which does not converge in norm, and let E be an isomorphically precisely norming set for X such that (*) ∑_n |x*(x_{n+1} - x_n)| < ∞, ∀x* ∈ E. Then there exists a subsequence of (x_n) which spans an isomorphically polyhedral Banach space. It follows immediately from results of V. Fonf that the converse is also true: If Y is a separable isomorphically polyhedral Banach space then there exists a normalized M-basis (x_n) which spans Y and there exists an isomorphically precisely norming set E for Y such that (*) is satisfied. As an application of this subsequence characterization of sequences spanning isomorphically polyhedral Banach spaces we obtain a strengthening of a result of J. Elton, and an Orlicz-Pettis type result.
LA - eng
KW - isomorphically polyhedral Banach space; normalized M-basis; Orlicz-Pettis type result
UR - http://eudml.org/doc/216460
ER -

References

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  13. [KP] M. I. Kadets and A. Pełczyński, Basic sequences, biorthogonal systems and norming sets in Banach and Fréchet spaces, Studia Math. 25 (1965), 297-323. Zbl0135.34504
  14. [K] V. L. Klee, Polyhedral sections of convex bodies, Acta Math. 4 (1966), 235-242. 
  15. [L] D. Leung, Some isomorphically polyhedral Orlicz sequence spaces, Israel J. Math. 87 (1994), 117-128. Zbl0804.46014
  16. [M] V. D. Milman, Geometric theory of Banach spaces, I, The theory of bases and minimal systems, Russian Math. Surveys 25 (3) (1970), 111-170. 
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