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1
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Modulus of dentability in $L¹ + L^{∞}$

100%
Bohonos A., Płuciennik R.
Banach Center Publications
|
2008
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tom 79
|
nr 1
39-51
EN
We introduce the notion of the modulus of dentability defined for any point of the unit sphere S(X) of a Banach space X. We calculate effectively this modulus for denting points of the unit ball of the classical interpolation space $L¹ + L^{∞}.$ Moreover, a criterion for denting points of the unit ball in this space is given. We also show that none of denting points of the unit ball of $L¹ + L^{∞}$ is a LUR-point. Consequently, the set of LUR-points of the unit ball of $L¹ + L^{∞}$ is empty.
2
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A strongly extreme point need not be a denting point in Orlicz spaces equipped with the Orlicz norm

100%
Bohonos A., Płuciennik R.
Banach Center Publications
|
2011
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tom 92
|
nr 1
37-44
EN
There are necessary conditions for a point x from the unit sphere to be a denting point of the unit ball of Orlicz spaces equipped with the Orlicz norm generated by arbitrary Orlicz functions. In contrast to results in [12, 17, 16], we present also examples of Orlicz spaces in which strongly extreme points of the unit ball are not denting points.
3
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Uniform \(\lambda\)-property in \(L^1\cap L^\infty\)

100%
Bohonos A., Płuciennik R.
Commentationes Mathematicae
|
2015
|
tom 55
|
nr 2
EN
Here it is proved that the space \(L^{1}\cap L^{\infty }\) equipped with the standard interpolation norm \(\left\Vert \cdot \right\Vert _{L^{1}\cap L^{\infty }}=\max \left\{ \left\Vert \cdot \right\Vert _{L^{1}},\left\Vert \cdot \right\Vert _{L^{\infty }}\right\} \) has the uniform \(\lambda \)-property if and only if \(\mu (T)\leq 1.\) Replacing the standard norm with an equivalent one \(\left\Vert \cdot \right\Vert _{L^{1}\cap L^{\infty }}^{\prime }= \) \(\left\Vert \cdot \right\Vert _{L^{1}}+\left\Vert \cdot \right\Vert _{L^{\infty }}\), a different result is obtained.: \((L^{1}\cap L^{\infty }, \left\Vert \cdot \right\Vert _{L^{1}\cap L^{\infty }}^{\prime } )\) has the uniform \(\lambda \)-property if and only if \(\mu (T)
4
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Banach-Saks property in some Banach sequence spaces

81%
Cui Y., Hudzik H., Płuciennik R.
Annales Polonici Mathematici
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1996-1997
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tom 65
|
nr 2
193-202
EN
It is proved that for any Banach space X property (β) defined by Rolewicz in [22] implies that both X and X* have the Banach-Saks property. Moreover, in Musielak-Orlicz sequence spaces, criteria for the Banach-Saks property, the near uniform convexity, the uniform Kadec-Klee property and property (H) are given.
5
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Errata to the paper "Banach-Saks property in some Banach sequence spaces" (Ann. Polon. Math. 65 (1997), 193-202)

81%
Cui Y., Hudzik H., Płuciennik R.
Annales Polonici Mathematici
|
1996-1997
|
tom 65
|
nr 3
303
6
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A note on strict K-monotonicity of some symmetric function spaces

81%
Ciesielski M., Kolwicz P., Płuciennik R.
Commentationes Mathematicae
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2013
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tom 53
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nr 2
EN
We discuss some sufficient and necessary conditions for strict K-monotonicity of some important concrete symmetric spaces. The criterion for strict monotonicity of the Lorentz space \(\Gamma _{p,w}\) with \(0\)
7
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A generalized projection decomposition in Orlicz-Bochner spaces

81%
Hudzik H., Płuciennik R., Wang Y.
Banach Center Publications
|
2005
|
tom 68
|
nr 1
61-70
EN
In this paper, a precise projection decomposition in reflexive, smooth and strictly convex Orlicz-Bochner spaces is given by the representation of the duality mapping. As an application, a representation of the metric projection operator on a closed hyperplane is presented.
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