luxemburg norm
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2021 ◽  
Vol 15 (3) ◽  
Author(s):  
Xiaoyan Li ◽  
Yunan Cui ◽  
Marek Wisla

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.



2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Shaoyong Zhang ◽  
Meiling Zhang ◽  
Yujia Zhan

It is well known that the modulus of nearly uniform smoothness related with the fixed point property is important in Banach spaces. In this paper, we prove that the modulus of nearly uniform smoothness in Köthe sequence spaces without absolutely continuous norm is ΓX(t)=t. Meanwhile, the formula of the modulus of nearly uniform smoothness in Orlicz sequence spaces equipped with the Luxemburg norm is given. As a corollary, we get a criterion for nearly uniform smoothness of Orlicz sequence spaces equipped with the Luxemburg norm. Finally, the equivalent conditions of R(a,l(Φ))<1+a and RW(a,l(Φ))<1+a are given.



2019 ◽  
Vol 10 (1) ◽  
pp. 81-96
Author(s):  
Wanzhong Gong ◽  
Xiaoli Dong ◽  
Kangji Wang


2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Vatan Karakaya ◽  
Fatma Altun

We introduce a new sequence space which is defined by the operatorW=(wnk)on the sequence spaceℓ(p). We define a modular functional on this space and investigate structure of this space equipped with Luxemburg norm. Also we study some geometric properties which are called Kadec-Klee, k-NUC, and uniform Opial properties and prove that this new space possesses these properties.





2012 ◽  
Vol 43 (2) ◽  
pp. 159-170
Author(s):  
Mehmet Sengonul

In this paper, using a modular, we have defined the modular space $M_{m^*}(p)$ and we have shown that the sequence space $M_{m^*}(p)$ equipped with the Luxemburg norm is rotund and possesses H-property (or Kadec-Klee property) when $p=(p_k)$ is bounded with $p_k>1$ for all $k\in\mathbb{N}$.



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