# Banach spaces with a supershrinking basis

Studia Mathematica (1999)

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

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topLópez, 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

top- [1] J. Diestel, Sequences and Series in Banach Spaces, Springer, 1984.
- [2] N. Ghoussoub and B. Maurey, ${G}_{\delta}$-embeddings in Hilbert space, J. Funct. Anal. 61 (1985), 72-97. Zbl0565.46011
- [3] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Ergeb. Math. Grenzgeb. 92, Springer, 1977. Zbl0362.46013
- [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] H. Rosenthal, A subsequence principle characterizing Banach spaces containing ${c}_{0}$, Bull. Amer. Math. Soc. 30 (1994), 227-233.
- [6] H. Rosenthal, Boundedly complete weak-Cauchy sequences in Banach spaces, preprint.

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