Uniform Gateaux differentiability and weak uniform rotundity in Musielak–Orlicz function spaces of Bochner type equipped with the Luxemburg norm

2012 ◽  
Vol 75 (6) ◽  
pp. 3009-3020 ◽  
Author(s):  
Shaoqiang Shang ◽  
Yunan Cui ◽  
Henryk Hudzik
Keyword(s):  
Orlicz Function ◽  
Function Spaces ◽  
Luxemburg Norm ◽  
Gâteaux Differentiability ◽  
Uniform Rotundity ◽  
Gateaux Differentiability

Uniform Gateaux differentiability and weak uniform rotundity of Musielak–Orlicz function spaces

2004 ◽  
Vol 56 (8) ◽  
pp. 1133-1149 ◽  
Author(s):  
Liu Li Fang ◽  
Wang Ting Fu ◽  
Henryk Hudzik
Keyword(s):  
Orlicz Function ◽  
Function Spaces ◽  
Gâteaux Differentiability ◽  
Uniform Rotundity ◽  
Gateaux Differentiability

Uniform rotundity of Orlicz function spaces equipped with the p-Amemiya norm

2018 ◽  
Vol 291 (10) ◽  
pp. 1514-1532 ◽  
Author(s):  
Radosław Kaczmarek
Keyword(s):  
Orlicz Function ◽  
Function Spaces ◽  
Uniform Rotundity ◽  
Amemiya Norm

scholarly journals On the weakly strongly exposed property and some smoothness properties of Orlicz spaces

1996 ◽  
Vol 54 (3) ◽  
pp. 431-440
Author(s):  
Yunan Cui ◽  
Henry K. Hudzik ◽  
Hongwei Zhu
Keyword(s):  
Banach Space ◽  
Orlicz Function ◽  
Orlicz Spaces ◽  
Function Spaces ◽  
Luxemburg Norm ◽  
Orlicz Norm ◽  
Dual Property ◽  
Order Continuous

The notion of a weakly strongly exposed Banach space is introduced and it is shown that this property is the dual property of very smoothness. Criteria for this property in Orlicz function spaces equipped with the Orlicz norm are presented. Criteria for strong smoothness and very smoothness of their subspaces of order continuous elements in the case of the Luxemburg norm are also given.

scholarly journals Uniform rotundity in every direction of Orlicz function spaces equipped with the p-Amemiya norm

2018 ◽  
Vol 70 (1) ◽  
pp. 71-86
Author(s):  
Radosław Kaczmarek
Keyword(s):  
Orlicz Function ◽  
Function Spaces ◽  
Uniform Rotundity ◽  
Amemiya Norm

Points of monotonicity in Musielak-Orlicz function spaces endowed with the Luxemburg norm

2004 ◽  
Vol 82 (6) ◽  
Author(s):  
H. Hudzik ◽  
X. B. Liu ◽  
T. F. Wang
Keyword(s):  
Orlicz Function ◽  
Function Spaces ◽  
Luxemburg Norm

Γ‎-Null and Γ‎N-Null Sets

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Finite Dimension ◽  
Baire Category ◽  
Borel Sets ◽  
Basic Properties ◽  
Gâteaux Differentiability ◽  
Gateaux Differentiability ◽  
Null Set

This chapter introduces the notions of Γ‎-null and Γ‎ₙ-null sets, which are σ‎-ideals of subsets of a Banach space X. Γ‎-null set is key for the strongest known general Fréchet differentiability results in Banach spaces, whereas Γ‎ₙ-null set presents a new, more refined concept. The reason for these notions comes from an (imprecise) observation that differentiability problems are governed by measure in finite dimension, but by Baire category when it comes to behavior at infinity. The chapter first relates Γ‎-null and Γ‎ₙ-null sets to Gâteaux differentiability before discussing their basic properties. It then describes Γ‎-null and Γ‎ₙ-null sets of low Borel classes and presents equivalent definitions of Γ‎ₙ-null sets. Finally, it considers the separable determination of Γ‎-nullness for Borel sets.

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