# Mapping Properties of ${c}_{0}$

Colloquium Mathematicae (1999)

- Volume: 80, Issue: 2, page 235-244
- ISSN: 0010-1354

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topLewis, Paul. "Mapping Properties of $c_0$." Colloquium Mathematicae 80.2 (1999): 235-244. <http://eudml.org/doc/210714>.

@article{Lewis1999,

abstract = {Bessaga and Pełczyński showed that if $c_0$ embeds in the dual $X^*$ of a Banach space X, then $ℓ^1$ embeds as a complemented subspace of X. Pełczyński proved that every infinite-dimensional closed linear subspace of $ℓ^1$ contains a copy of $ℓ^1$ that is complemented in $ℓ^1$. Later, Kadec and Pełczyński proved that every non-reflexive closed linear subspace of $L^1 [0,1]$ contains a copy of $ℓ^1$ that is complemented in $L^1 [0,1]$. In this note a traditional sliding hump argument is used to establish a simple mapping property of $c_0$ which simultaneously yields extensions of the preceding theorems as corollaries. Additional classical mapping properties of $c_0$ are briefly discussed and applications are given.},

author = {Lewis, Paul},

journal = {Colloquium Mathematicae},

keywords = {sliding hump argument; mapping property; complemented subspace},

language = {eng},

number = {2},

pages = {235-244},

title = {Mapping Properties of $c_0$},

url = {http://eudml.org/doc/210714},

volume = {80},

year = {1999},

}

TY - JOUR

AU - Lewis, Paul

TI - Mapping Properties of $c_0$

JO - Colloquium Mathematicae

PY - 1999

VL - 80

IS - 2

SP - 235

EP - 244

AB - Bessaga and Pełczyński showed that if $c_0$ embeds in the dual $X^*$ of a Banach space X, then $ℓ^1$ embeds as a complemented subspace of X. Pełczyński proved that every infinite-dimensional closed linear subspace of $ℓ^1$ contains a copy of $ℓ^1$ that is complemented in $ℓ^1$. Later, Kadec and Pełczyński proved that every non-reflexive closed linear subspace of $L^1 [0,1]$ contains a copy of $ℓ^1$ that is complemented in $L^1 [0,1]$. In this note a traditional sliding hump argument is used to establish a simple mapping property of $c_0$ which simultaneously yields extensions of the preceding theorems as corollaries. Additional classical mapping properties of $c_0$ are briefly discussed and applications are given.

LA - eng

KW - sliding hump argument; mapping property; complemented subspace

UR - http://eudml.org/doc/210714

ER -

## References

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- [11] H. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math. 37 (1970), 13-36. Zbl0227.46027
- [12] E. Saab and P. Saab, On complemented copies of ${c}_{0}$ in injective tensor products, in: Contemp. Math. 52, Amer. Math. Soc., 1986, 131-135. Zbl0589.46057
- [13] I. Singer, Bases in Banach Spaces II, Springer, Berlin, 1981.
- [14] A. Sobczyk, Projections of the space m on its subspace c, Bull. Amer. Math. Soc. 47 (1941), 78-106. Zbl67.0403.03
- [15] W. Veech, Short proof of Sobczyk's theorem, Proc. Amer. Math. Soc. 28 (1971), 627-628. Zbl0213.39402

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