Smooth, very smooth and strongly smooth points in Musielak-Orlicz sequence spaces

Bian, Shurong; Hudzik, Henryk; Wang, Tingfu

doi:10.1017/s0004972700019523

Cited by 9 publications

(3 citation statements)

References 10 publications

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“…We have just proven analogues of Lemmas 0.1, 0.2, 0.3 and 0.4 from [2]. On the base of these results necessary and sufficient conditions for smooth points of the Musielak-Orlicz space were given in that article.…”

Section: Smooth Points Of L ϕ (γ)mentioning

confidence: 85%

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Basic Topological and Geometric Properties of Orlicz Spaces over an Arbitrary Set of Atoms

Hudzik¹,

Szymaszkiewicz²

2008

Z. Anal. Anwend.

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Orlicz spaces l ϕ (Γ) over an arbitrary set Γ, being a natural generalizations of Orlicz sequence spaces are studied. The following problems in these spaces are considered: relationships between the Luxemburg norm and the modular, Fatou property, relationships between the Luxemburg norm and the Orlicz norm, equality of the Orlicz norm and the Amemiya norm, order continuous elements, a formula for the norm in the quotient space l ϕ (Γ)/h ϕ (Γ) in terms of the modular I ϕ for both the Luxemburg and the Orlicz norm, the problem when the equality of the space l ϕ (Γ) and its subspace h ϕ (Γ) holds, isometric representation of the dual spaces (h ϕ (Γ)) * , (l ϕ (Γ)) * , (h ϕ o (Γ)) * and (l ϕ o (Γ)) * , representation of support functionals, criteria for smooth points and extreme points of S(l ϕ (Γ)) and problem of the existence of such points. It is worthy noticing that the problem of the existence of smooth points on S(l ϕ (Γ)) depends essentially on the assumption if Γ is countable or not.

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Section: Smooth Points Of L ϕ (γ)mentioning

confidence: 85%

“…Next we will point out the differences between the spaces depending on the cardinality of the set Γ. Smooth points in Orlicz spaces or in Musielak-Orlicz spaces corresponding to σ-finite measure spaces were investigated among others in [2][3][4][5][6][9][10][11].…”

Section: Smooth Points Of L ϕ (γ)mentioning

confidence: 99%

Basic Topological and Geometric Properties of Orlicz Spaces over an Arbitrary Set of Atoms

Hudzik¹,

Szymaszkiewicz²

2008

Z. Anal. Anwend.

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“…Weyssenhoffa 11, 85-072 Bydgoszcz, Poland e-mail: mwojt@ukw.edu.pl Let us recall that an element x ∈ X \{0} is said to be smooth if the set Grad(x) is a singleton. Over the last 30 years, there have been published many papers devoted to smooth points in concrete Banach lattices (see, e.g., [4,[7][8][9] and the references therein). Most of the proofs are too technical, and there is no general point of view both on the structure of Grad(x) and smooth points of X .…”

Section: Wójtowicz (B)mentioning

confidence: 99%

The positive aspects of smoothness in Banach lattices

Wójtowicz

2012

Positivity

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Let X be a Banach lattice, and let x ∈ X \{0}. We study the structure of the set Grad(x), of all supporting functionals of x. If X is a Dedekind σ -complete Banach lattice, there is an isometry from Grad(x) onto Grad(|x|); hence the elements x and |x| are smooth simultaneously. And if, additionally, X * is strictly monotone then Grad(|x|) consists of positive functionals. As a by-product of our results we obtain that an arbitrary Banach lattice X is strictly monotone whenever its dual X * is smooth.

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Contractive and optimal sets in modular spaces

Kamińska

Lewicki

2004

Mathematische Nachrichten

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We study contractive, existence and optimal sets in modular spaces. We define the analogous classes of sets with respect to a modular and generalize a number of results due to Beauzamy and Maurey [3] and Bruck [9] to modular spaces. We then apply them to show that in certain Köthe sequence spaces all of these notions are equivalent, extending a result of Davis and Enflo [11] for lp spaces, 1 < p < ∞. We also show that in class of Orlicz sequence spaces the theorem that any optimal set in lp spaces, 1 < p < ∞, is an intersection of half-spaces determined by one-complemented hyperplanes in lp, is no longer true. We then find a class of optimal sets, called strongly ρ-optimal, such that this characterization holds true in Musielak-Orlicz sequence spaces. We provide examples which show that all families of sets discussed in the paper like optimal, existence or contractive with respect to a norm or a modular or their strong counterparts, are not equivalent in general.

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Smooth, very smooth and strongly smooth points in Musielak-Orlicz sequence spaces

Abstract: Criteria for smooth points, very smooth points and strongly smooth points in Musielak-Orlicz sequence spaces equipped with the Luxemburg norm are given.

Cited by 9 publications

References 10 publications

Basic Topological and Geometric Properties of Orlicz Spaces over an Arbitrary Set of Atoms

Basic Topological and Geometric Properties of Orlicz Spaces over an Arbitrary Set of Atoms

The positive aspects of smoothness in Banach lattices

Contractive and optimal sets in modular spaces

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