# Article

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Keywords:
Musielak-Orlicz sequence space; copy of $l^1$; copy of $c_o$
Summary:
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.
References:
[A] Alherk G.: On the non-$l_n^{(1)}$ and locally uniformly non-$l_n^{(1)}$ properties and $l^1$ copies in Musielak-Orlicz space. Comment. Math. Univ. Carolinae 31 (1990), 435-443. MR 1078478
[AB] Aliprantis C.D., Burkinshaw O.: Positive operators. Pure and Applied Math., Academic Press, Inc., 1985. MR 0809372 | Zbl 1098.47001
[BP] Bessaga C., Pełczyński A.: On bases and unconditional convergence of series in Banach spaces. Studia Math. 17 (1958), 151-164. MR 0115069
[DH] Denker M., Hudzik H.: Uniformly non-$l^{(1)}_n$ Musielak-Orlicz sequence spaces. Proc. Indian Acad. Sci. 101 (1991), 71-86. MR 1125480
[H1] Hudzik H.: On some equivalent conditions in Musielak-Orlicz spaces. Comment. Math. (Prace Matem.) 24 (1984), 57-64. MR 0759055 | Zbl 0564.46022
[H2] Hudzik H.: Orlicz spaces containing a copy of $l^1$. Math. Japonica 34 (1989), 747-759. MR 1022152
[HY] Hudzik H., Ye Y.: Support functionals and smoothness in Musielak-Orlicz sequence spaces endowed with the Luxemburg norm. Comment. Math. Univ. Carolinae 31 (1990), 661-684. MR 1091364 | Zbl 0721.46012
[K] Kamińska A.: Flat Orlicz-Musielak sequence spaces. Bull. Acad. Polon. Sci. Math. 30 (1982), 347-352. MR 0707748
[KA] Kantorovich L.V., Akilov G.P.: Functional Analysis (in Russian). Nauka, Moscow, 1977. MR 0511615
[KR] Krasnoselskii M.A., Rutickii Ya.B.: Convex functions and Orlicz spaces. Groningen, 1961 (translation).
[L] Luxemburg W.A.J.: Banach function spaces. Thesis, Delft, 1955. MR 0072440 | Zbl 0162.44701
[M] Musielak J.: Orlicz spaces and modular spaces. Lecture Notes in Math. 1034, SpringerVerlag, 1983. MR 0724434 | Zbl 0557.46020
[RR] Rao M.M., Ren Z.D.: Theory of Orlicz spaces. Pure and Applied Mathematics, Marcel Dekker, 1991. MR 1113700 | Zbl 0724.46032
[T] Turett B.: Fenchel-Orlicz spaces. Dissertationes Math. 181 (1980), 1-60. MR 0578390 | Zbl 0435.46025

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