scholarly journals Strongly Extreme Points and Middle Point Locally Uniformly Convex in Orlicz Spaces Equipped with s-Norm

2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Yunan Cui ◽  
Yujia Zhan
Keyword(s):  
Banach Spaces ◽  
Orlicz Space ◽  
Necessary Conditions ◽  
Orlicz Function ◽  
Orlicz Spaces ◽  
Extreme Points ◽  
Uniformly Convex ◽  
Middle Point ◽  
Sufficient And Necessary Conditions

As is well known, the extreme points and strongly extreme points play important roles in Banach spaces. In this paper, the criterion for strongly extreme points in Orlicz spaces equipped with s-norm is given. We complete solved criterion-Orlicz space that generated by Orlicz function. And the sufficient and necessary conditions for middle point locally uniformly convex in Orlicz spaces equipped with s-norm are obtained.

Orlicz Spaces Without Strongly Extreme Points and Without H-Points

1993 ◽  
Vol 36 (2) ◽  
pp. 173-177 ◽  
Author(s):  
Henryk Hudzik
Keyword(s):  
Extreme Point ◽  
Orlicz Space ◽  
Unit Sphere ◽  
Orlicz Function ◽  
Orlicz Spaces ◽  
Extreme Points ◽  
Luxemburg Norm ◽  
Nonatomic Measure

AbstractW. Kurc [5] has proved that in the unit sphere of Orlicz space LΦ(μ) generated by an Orlicz function Φ satisfying the suitable Δ2-condition and equipped with the Luxemburg norm every extreme point is strongly extreme. In this paper it is proved in the case of a nonatomic measure μ that the unit sphere of the Orlicz space LΦ(μ) generated by an Orlicz function Φ which does not satisfy the suitable Δ2-condition and equipped with the Luxemburg norm has no strongly extreme point and no H-point.

scholarly journals Smoothness of Orlicz function spaces equipped with the p-Amemiya norm

2021 ◽  
Vol 15 (3) ◽  
Author(s):  
Xiaoyan Li ◽  
Yunan Cui ◽  
Marek Wisla
Keyword(s):  
Explicit Form ◽  
Necessary Conditions ◽  
Orlicz Function ◽  
Orlicz Spaces ◽  
Luxemburg Norm ◽  
Orlicz Norm ◽  
Dual Norm ◽  
Sufficient And Necessary Conditions ◽  
Supporting Functional ◽  
Amemiya Norm

AbstractIn this paper, we will use the convex modular $$\rho ^{*}(f)$$ ρ ∗ ( f ) to investigate $$\Vert f\Vert _{\Psi ,q}^{*}$$ ‖ f ‖ Ψ , q ∗ on $$(L_{\Phi })^{*}$$ ( L Φ ) ∗ defined by the formula $$\Vert f\Vert _{\Psi ,q}^{*}=\inf _{k>0}\frac{1}{k}s_{q}(\rho ^{*}(kf))$$ ‖ f ‖ Ψ , q ∗ = inf k > 0 1 k s q ( ρ ∗ ( k f ) ) , which is the norm formula in Orlicz dual spaces equipped with p-Amemiya norm. The attainable points of dual norm $$\Vert f\Vert _{\Psi ,q}^{*}$$ ‖ f ‖ Ψ , q ∗ are discussed, the interval for dual norm $$\Vert f\Vert _{\Psi ,q}^{*}$$ ‖ f ‖ Ψ , q ∗ attainability is described. By presenting the explicit form of supporting functional, we get sufficient and necessary conditions for smooth points. As a result, criteria for smoothness of $$L_{\Phi ,p}~(1\le p\le \infty )$$ L Φ , p ( 1 ≤ p ≤ ∞ ) is also obtained. The obtained results unify, complete and extended as well the results presented by a number of paper devoted to studying the smoothness of Orlicz spaces endowed with the Luxemburg norm and the Orlicz norm separately.

Extreme and Exposed Points in Orlicz Spaces

1992 ◽  
Vol 44 (3) ◽  
pp. 505-515 ◽  
Author(s):  
R. Grzaślewicz ◽  
H. Hudzik ◽  
W. Kurc
Keyword(s):  
Orlicz Space ◽  
Unit Sphere ◽  
Measure Space ◽  
Orlicz Function ◽  
Orlicz Spaces ◽  
Extreme Points ◽  
Infinite Measure ◽  
Nonatomic Measure

AbstractExtreme points of the unit sphere in any Orlicz space over a measure space that contains no atoms of infinite measure are characterized. In the case of a finite-valued Orlicz function and a nonatomic measure space, exposed points of the unit sphere in these spaces are characterized too. Some corollaries and examples are also given.

scholarly journals Uniformly Nonsquare in Orlicz Space Equipped with the Mazur-Orlicz F -Norm

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Yunan Cui ◽  
Tongyu Wang
Keyword(s):  
Banach Spaces ◽  
Orlicz Space ◽  
Orlicz Spaces ◽  
Normed Spaces ◽  
Definition Of

The definition of uniformly nonsquareness in Banach spaces is extended to F -normed spaces. Most of the results from this paper concern (uniformly) nonsquareness in the sense of James or in the sense of Schäffer in Orlicz spaces equipped with the Mazur-Orlicz F -norm. It is well known that uniform nonsquareness in the sense of Schäffer and in the sense of James are equivalent in Banach spaces. In this paper, we found that uniform nonsquareness in the sense of James and in the sense of Schäffer are not equivalent for F -normed spaces. Criteria for Orlicz spaces equipped with the Mazur-Orlicz F -norm to be nonsquare and uniformly nonsquare in the sense of James or in the sense of Schäffer are given.

scholarly journals Extreme points and rotundity of Orlicz-Sobolev spaces

2002 ◽  
Vol 32 (5) ◽  
pp. 293-299
Author(s):  
Shutao Chen ◽  
Changying Hu ◽  
Charles Xuejin Zhao
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Sobolev Spaces ◽  
Necessary Conditions ◽  
Nonlinear Partial Differential Equations ◽  
Extreme Points ◽  
Necessary Condition ◽  
Partial Differential ◽  
Sufficient And Necessary Conditions ◽  
Sufficient And Necessary Condition

It is well known that Sobolev spaces have played essential roles in solving nonlinear partial differential equations. Orlicz-Sobolev spaces are generalized from Sobolev spaces. In this paper, we present sufficient and necessary conditions of extreme points of Orlicz-Sobolev spaces. A sufficient and necessary condition of rotundity of Orlicz-Sobolev spaces is obtained.

scholarly journals Third Version of Weak Orlicz–Morrey Spaces and Its In-clusion Properties

2019 ◽  
Vol 4 (2) ◽  
pp. 257-262
Author(s):  
Al Azhary Masta ◽  
Siti Fatimah ◽  
Muhammad Taqiyuddin
Keyword(s):  
Morrey Space ◽  
Necessary Conditions ◽  
Orlicz Spaces ◽  
Morrey Spaces ◽  
The Third ◽  
Inclusion Relations ◽  
Sufficient And Necessary Conditions ◽  
Weak Morrey Spaces

Orlicz–Morrey spaces are generalizations of Orlicz spaces and Morrey spaces which were first introduced by Nakai. There are  three  versions  of  Orlicz–Morrey  spaces.  In  this  article,  we discussed  the  third  version  of  weak  Orlicz–Morrey  space, which is an enlargement of third version of (strong) Orlicz– Morrey space. Similar to its first version and second version, the third version of weak Orlicz-Morrey space is considered as  a  generalization  of  weak  Orlicz  spaces,  weak  Morrey spaces,  and  generalized  weak  Morrey  spaces.  This  study investigated  some  properties  of the third  version of weak Orlicz–Morrey spaces, especially the sufficient and necessary conditions for inclusion relations between two these spaces. One of the keys to get our result is to estimate the quasi- norm of characteristics function of open balls in ℝ.

scholarly journals On projection constant problems and the existence of metric projections in normed spaces

2001 ◽  
Vol 6 (7) ◽  
pp. 401-411 ◽  
Author(s):  
Entisarat El-Shobaky ◽  
Sahar Mohammed Ali ◽  
Wataru Takahashi
Keyword(s):  
Banach Spaces ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Metric Projection ◽  
Infinite Matrix ◽  
Normed Spaces ◽  
Reflexive Banach Spaces ◽  
Linear Spaces ◽  
Sufficient And Necessary Conditions ◽  
Projection Constant

We give the sufficient conditions for the existence of a metric projection onto convex closed subsets of normed linear spaces which are reduced conditions than that in the case of reflexive Banach spaces and we find a general formula for the projections onto the maximal proper subspaces of the classical Banach spacesl p,1≤p<∞andc 0. We also give the sufficient and necessary conditions for an infinite matrix to represent a projection operator froml p,1≤p<∞orc 0onto anyone of their maximal proper subspaces.

Almost transitivity of some function spaces

1994 ◽  
Vol 116 (3) ◽  
pp. 475-488 ◽  
Author(s):  
Peter Greim ◽  
James E. Jamison ◽  
Anna Kamińska
Keyword(s):  
Unit Ball ◽  
Orlicz Space ◽  
Orlicz Spaces ◽  
Extreme Points ◽  
Function Spaces ◽  
Separable Spaces

AbstractThe almost transitive norm problem is studied for Lp (μ, X), C(K, X) and for certain Orlicz and Musielak-Orlicz spaces. For example if p ≠ 2 < ∞ then Lp (μ) has almost transitive norm if and only if the measure μ is homogeneous. It is shown that the only Musielak-Orlicz space with almost transitive norm is the Lp-space. Furthermore, an Orlicz space has an almost transitive norm if and only if the norm is maximal. Lp (μ, X) has almost transitive norm if Lp(μ) and X have. Separable spaces with non-trivial Lp-structure fail to have transitive norms. Spaces with nontrivial centralizers and extreme points in the unit ball also fail to have almost transitive norms.

scholarly journals A Note on Inclusion Properties of Weighted Orlicz Spaces

2020 ◽  
Vol 26 (1) ◽  
pp. 128-136
Author(s):  
Al Azhary Masta ◽  
Ifronika ◽  
Muhammad Taqiyuddin
Keyword(s):  
Necessary Conditions ◽  
Orlicz Spaces ◽  
Inclusion Properties ◽  
Sufficient And Necessary Conditions

In this paper we present sufficient and necessary conditions for the inclusion relationbetween two weighted Orlicz spaces which complete the Osan\c{c}liol result in 2014.One of the keys to prove our results is to use the norm of the characteristic functionsof the balls in $\mathbb{R}^n$.

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