Copies of l 1 and c o in Musielak-Orlicz sequence spaces

Ghassan Alherk; Henryk Hudzik

Commentationes Mathematicae Universitatis Carolinae (1994)

  • Volume: 35, Issue: 1, page 9-19
  • ISSN: 0010-2628

Abstract

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Criteria in order that a Musielak-Orlicz sequence space l Φ contains an isomorphic as well as an isomorphically isometric copy of l 1 are given. Moreover, it is proved that if Φ = ( Φ i ) , where Φ i are defined on a Banach space, X does not satisfy the δ 2 o -condition, then the Musielak-Orlicz sequence space l Φ ( X ) of X -valued sequences contains an almost isometric copy of c o . In the case of X = I R it is proved also that if l Φ contains an isomorphic copy of c o , then Φ does not satisfy the δ 2 o -condition. These results extend some results of [A] and [H2] to Musielak-Orlicz sequence spaces.

How to cite

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Alherk, Ghassan, and Hudzik, Henryk. "Copies of $l^1$ and $c_o$ in Musielak-Orlicz sequence spaces." Commentationes Mathematicae Universitatis Carolinae 35.1 (1994): 9-19. <http://eudml.org/doc/247637>.

@article{Alherk1994,
abstract = {Criteria in order that a Musielak-Orlicz sequence space $l^\Phi $ contains an isomorphic as well as an isomorphically isometric copy of $l^1$ are given. Moreover, it is proved that if $\Phi = (\Phi _i)$, where $\Phi _i$ are defined on a Banach space, $X$ does not satisfy the $\delta ^o_2$-condition, then the Musielak-Orlicz sequence space $l^\Phi (X)$ of $X$-valued sequences contains an almost isometric copy of $c_o$. In the case of $X = I\!\!R$ it is proved also that if $l^\Phi $ contains an isomorphic copy of $c_o$, then $\Phi $ does not satisfy the $\delta ^o_2$-condition. These results extend some results of [A] and [H2] to Musielak-Orlicz sequence spaces.},
author = {Alherk, Ghassan, Hudzik, Henryk},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Musielak-Orlicz sequence space; copy of $l^1$; copy of $c_o$; Musielak-Orlicz sequence space; -valued sequences},
language = {eng},
number = {1},
pages = {9-19},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Copies of $l^1$ and $c_o$ in Musielak-Orlicz sequence spaces},
url = {http://eudml.org/doc/247637},
volume = {35},
year = {1994},
}

TY - JOUR
AU - Alherk, Ghassan
AU - Hudzik, Henryk
TI - Copies of $l^1$ and $c_o$ in Musielak-Orlicz sequence spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1994
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 35
IS - 1
SP - 9
EP - 19
AB - Criteria in order that a Musielak-Orlicz sequence space $l^\Phi $ contains an isomorphic as well as an isomorphically isometric copy of $l^1$ are given. Moreover, it is proved that if $\Phi = (\Phi _i)$, where $\Phi _i$ are defined on a Banach space, $X$ does not satisfy the $\delta ^o_2$-condition, then the Musielak-Orlicz sequence space $l^\Phi (X)$ of $X$-valued sequences contains an almost isometric copy of $c_o$. In the case of $X = I\!\!R$ it is proved also that if $l^\Phi $ contains an isomorphic copy of $c_o$, then $\Phi $ does not satisfy the $\delta ^o_2$-condition. These results extend some results of [A] and [H2] to Musielak-Orlicz sequence spaces.
LA - eng
KW - Musielak-Orlicz sequence space; copy of $l^1$; copy of $c_o$; Musielak-Orlicz sequence space; -valued sequences
UR - http://eudml.org/doc/247637
ER -

References

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  8. Kamińska A., Flat Orlicz-Musielak sequence spaces, Bull. Acad. Polon. Sci. Math. 30 (1982), 347-352. (1982) MR0707748
  9. Kantorovich L.V., Akilov G.P., Functional Analysis (in Russian), Nauka, Moscow, 1977. MR0511615
  10. Krasnoselskii M.A., Rutickii Ya.B., Convex functions and Orlicz spaces, Groningen, 1961 (translation). 
  11. Luxemburg W.A.J., Banach function spaces, Thesis, Delft, 1955. Zbl0162.44701MR0072440
  12. Musielak J., Orlicz spaces and modular spaces, Lecture Notes in Math. 1034, SpringerVerlag, 1983. Zbl0557.46020MR0724434
  13. Rao M.M., Ren Z.D., Theory of Orlicz spaces, Pure and Applied Mathematics, Marcel Dekker, 1991. Zbl0724.46032MR1113700
  14. Turett B., Fenchel-Orlicz spaces, Dissertationes Math. 181 (1980), 1-60. (1980) Zbl0435.46025MR0578390

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