On the complemented subspaces of the Schreier spaces

I. Gasparis; D. Leung

Studia Mathematica (2000)

  • Volume: 141, Issue: 3, page 273-300
  • ISSN: 0039-3223

Abstract

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It is shown that for every 1 ≤ ξ < ω, two subspaces of the Schreier space X ξ generated by subsequences ( e l n ξ ) and ( e m n ξ ) , respectively, of the natural Schauder basis ( e n ξ ) of X ξ are isomorphic if and only if ( e l n ξ ) and ( e m n ξ ) are equivalent. Further, X ξ admits a continuum of mutually incomparable complemented subspaces spanned by subsequences of ( e n ξ ) . It is also shown that there exists a complemented subspace spanned by a block basis of ( e n ξ ) , which is not isomorphic to a subspace generated by a subsequence of ( e n ζ ) , for every 0 ζ ξ . Finally, an example is given of an uncomplemented subspace of X ξ which is spanned by a block basis of ( e n ξ ) .

How to cite

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Gasparis, I., and Leung, D.. "On the complemented subspaces of the Schreier spaces." Studia Mathematica 141.3 (2000): 273-300. <http://eudml.org/doc/216785>.

@article{Gasparis2000,
abstract = {It is shown that for every 1 ≤ ξ < ω, two subspaces of the Schreier space $X^ξ$ generated by subsequences $(e_\{l_n\}^\{ξ\})$ and $(e_\{m_n\}^\{ξ\})$, respectively, of the natural Schauder basis $(e_\{n\}^\{ξ\})$ of $X^ξ$ are isomorphic if and only if $(e_\{l_n\}^\{ξ\})$ and $(e_\{m_n\}^\{ξ\})$ are equivalent. Further, $X^ξ$ admits a continuum of mutually incomparable complemented subspaces spanned by subsequences of $(e_\{n\}^\{ξ\})$. It is also shown that there exists a complemented subspace spanned by a block basis of $(e_\{n\}^\{ξ\})$, which is not isomorphic to a subspace generated by a subsequence of $(e_n^ζ)$, for every $0 ≤ ζ ≤ ξ$. Finally, an example is given of an uncomplemented subspace of $X^ξ$ which is spanned by a block basis of $(e_\{n\}^\{ξ\})$.},
author = {Gasparis, I., Leung, D.},
journal = {Studia Mathematica},
keywords = {Schreier sets; complemented subspace},
language = {eng},
number = {3},
pages = {273-300},
title = {On the complemented subspaces of the Schreier spaces},
url = {http://eudml.org/doc/216785},
volume = {141},
year = {2000},
}

TY - JOUR
AU - Gasparis, I.
AU - Leung, D.
TI - On the complemented subspaces of the Schreier spaces
JO - Studia Mathematica
PY - 2000
VL - 141
IS - 3
SP - 273
EP - 300
AB - It is shown that for every 1 ≤ ξ < ω, two subspaces of the Schreier space $X^ξ$ generated by subsequences $(e_{l_n}^{ξ})$ and $(e_{m_n}^{ξ})$, respectively, of the natural Schauder basis $(e_{n}^{ξ})$ of $X^ξ$ are isomorphic if and only if $(e_{l_n}^{ξ})$ and $(e_{m_n}^{ξ})$ are equivalent. Further, $X^ξ$ admits a continuum of mutually incomparable complemented subspaces spanned by subsequences of $(e_{n}^{ξ})$. It is also shown that there exists a complemented subspace spanned by a block basis of $(e_{n}^{ξ})$, which is not isomorphic to a subspace generated by a subsequence of $(e_n^ζ)$, for every $0 ≤ ζ ≤ ξ$. Finally, an example is given of an uncomplemented subspace of $X^ξ$ which is spanned by a block basis of $(e_{n}^{ξ})$.
LA - eng
KW - Schreier sets; complemented subspace
UR - http://eudml.org/doc/216785
ER -

References

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