# On the complemented subspaces of the Schreier spaces

Studia Mathematica (2000)

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

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topGasparis, 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 -

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