Mapping Properties of c 0

Paul Lewis

Colloquium Mathematicae (1999)

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

Abstract

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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.

How to cite

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Lewis, 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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  1. [1] C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151-164. Zbl0084.09805
  2. [2] J. Brooks and N. Dinculeanu, Strong additivity, absolute continuity, and compactness in spaces of measures, J. Math. Anal. Appl. 45 (1974), 156-175. Zbl0284.28005
  3. [3] J. Diestel, Sequences and Series in Banach Spaces, Grad. Texts in Math. 92, Springer, New York, 1984. 
  4. [4] J. Diestel and J. J. Uhl, Jr., Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, 1977. 
  5. [5] N. Dunford, A mean ergodic theorem, Duke Math. J. 5 (1939), 635-646. 
  6. [6] N. Dunford and J. Schwartz, Linear Operators. Part I, Interscience, New York, 1958. 
  7. [7] M. I. Kadec and A. Pełczyński, Bases, lacunary sequences and complemented subspaces in the spaces L p , Studia Math. 21 (1962), 161-176. 
  8. [8] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer, Berlin, 1977. 
  9. [9] A. Pełczyński, Projections in certain Banach spaces, Studia Math. 19 (1960), 209-228. Zbl0104.08503
  10. [10] A. Pełczyński, On strictly singular and strictly cosingular operators. II, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 (1965), 37-41. Zbl0138.38604
  11. [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. [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. [13] I. Singer, Bases in Banach Spaces II, Springer, Berlin, 1981. 
  14. [14] A. Sobczyk, Projections of the space m on its subspace c, Bull. Amer. Math. Soc. 47 (1941), 78-106. Zbl67.0403.03
  15. [15] W. Veech, Short proof of Sobczyk's theorem, Proc. Amer. Math. Soc. 28 (1971), 627-628. Zbl0213.39402

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