Banach spaces with a supershrinking basis

Ginés López

Studia Mathematica (1999)

  • Volume: 132, Issue: 1, page 29-36
  • ISSN: 0039-3223

Abstract

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We prove that a Banach space X with a supershrinking basis (a special type of shrinking basis) without c 0 copies is somewhat reflexive (every infinite-dimensional subspace contains an infinite-dimensional reflexive subspace). Furthermore, applying the c 0 -theorem by Rosenthal, it is proved that X contains order-one quasireflexive subspaces if X is not reflexive. Also, we obtain a characterization of the usual basis in c 0 .

How to cite

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López, Ginés. "Banach spaces with a supershrinking basis." Studia Mathematica 132.1 (1999): 29-36. <http://eudml.org/doc/216584>.

@article{López1999,
abstract = {We prove that a Banach space X with a supershrinking basis (a special type of shrinking basis) without $c_0$ copies is somewhat reflexive (every infinite-dimensional subspace contains an infinite-dimensional reflexive subspace). Furthermore, applying the $c_0$-theorem by Rosenthal, it is proved that X contains order-one quasireflexive subspaces if X is not reflexive. Also, we obtain a characterization of the usual basis in $c_0$.},
author = {López, Ginés},
journal = {Studia Mathematica},
keywords = {quasireflexive Banach space; Banach space with a normalized and shrinking basis; reflexive subspaces; normalized supershrinking basis},
language = {eng},
number = {1},
pages = {29-36},
title = {Banach spaces with a supershrinking basis},
url = {http://eudml.org/doc/216584},
volume = {132},
year = {1999},
}

TY - JOUR
AU - López, Ginés
TI - Banach spaces with a supershrinking basis
JO - Studia Mathematica
PY - 1999
VL - 132
IS - 1
SP - 29
EP - 36
AB - We prove that a Banach space X with a supershrinking basis (a special type of shrinking basis) without $c_0$ copies is somewhat reflexive (every infinite-dimensional subspace contains an infinite-dimensional reflexive subspace). Furthermore, applying the $c_0$-theorem by Rosenthal, it is proved that X contains order-one quasireflexive subspaces if X is not reflexive. Also, we obtain a characterization of the usual basis in $c_0$.
LA - eng
KW - quasireflexive Banach space; Banach space with a normalized and shrinking basis; reflexive subspaces; normalized supershrinking basis
UR - http://eudml.org/doc/216584
ER -

References

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  1. [1] J. Diestel, Sequences and Series in Banach Spaces, Springer, 1984. 
  2. [2] N. Ghoussoub and B. Maurey, G δ -embeddings in Hilbert space, J. Funct. Anal. 61 (1985), 72-97. Zbl0565.46011
  3. [3] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Ergeb. Math. Grenzgeb. 92, Springer, 1977. Zbl0362.46013
  4. [4] G. López and J. F. Mena, RNP and KMP are equivalent for some Banach spaces with shrinking basis, Studia Math. 118 (1996), 11-17. Zbl0854.46016
  5. [5] H. Rosenthal, A subsequence principle characterizing Banach spaces containing c 0 , Bull. Amer. Math. Soc. 30 (1994), 227-233. 
  6. [6] H. Rosenthal, Boundedly complete weak-Cauchy sequences in Banach spaces, preprint. 

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