Orlicz Bochner Function Spaces Research Articles

I-convexity and Q-convexity of Orlicz-Bochner function spaces with the Luxemburg norm

I-convexity and Q-convexity of Orlicz-Bochner function spaces with the Luxemburg norm

I‐convexity and Q‐convexity in Orlicz–Bochner function spaces endowed with the Orlicz norm

AbstractSome characterizations of I‐convexity and Q‐convexity of Banach space are obtained. Moreover, the criteria is shown for Orlicz–Bochner function spaces endowed with the Orlicz norm being I‐convex as well as being Q‐convex.

I-convexity and Q-convexity in Orlicz–Bochner function spaces equipped with the Luxemburg norm

We study I-convexity and Q-convexity, two geometric properties introduced by Amir and Franchetti. We point out that a Banach space X has the weak fixed-point property when X is I-convex (or Q-convex) with a strongly bimonotone basis. By means of some characterizations of I-convexity and Q-convexity in Banach spaces, we obtain criteria for these two convexities in the Orlicz–Bochner function space L(M)(μ,X): that L(M)(μ,X) is I-convex (or Q-convex) if and only if L(M)(μ) is reflexive and X is I-convex (or Q-convex).

Musielak-Orlicz-Bochner function spaces which are uniformly noncreasy

In this paper, the criteria for uniform noncreasiness of Musielak–Orlicz–Bochner function spaces are given. Moreover authors also prove that the space LM(X) (resp LM0(X)) is uniformly noncreasy if and only if the space LM(X) (resp LM0(X)) is uniformly convex or uniformly smooth. As a corollary, the criteria for uniform noncreasiness of Musielak–Orlicz function spaces are given.

Uniformly non-[formula omitted], locally uniformly non-[formula omitted] and non-[formula omitted] properties in Orlicz–Bochner function spaces endowed with the Orlicz norm

Some criteria are given in this paper for uniformly non-ln(1), locally uniformly non-ln(1) and non-ln(1) properties of Orlicz–Bochner function spaces endowed with the Orlicz norm. As the consequence, we can immediately get the characterization of B-convexity, uniform nonsquareness, local uniform nonsquareness and nonsquareness in Orlicz–Bochner function spaces, although some of them were obtained in recent years.

Approximative compactness in Musielak–Orlicz function spaces of Bochner type

In this article, we give the criteria for approximative compactness of every proximinal convex subset of Musielak–Orlicz–Bochner function spaces equipped with the Orlicz norm. As a corollary, we give the criteria for approximative compactness of Musielak–Orlicz–Bochner function spaces equipped with the Orlicz norm.

Uniform nonsquareness and locally uniform nonsquareness in Orlicz–Bochner function spaces and applications

In this paper, criteria for uniform nonsquareness and locally uniform nonsquareness of Orlicz–Bochner function spaces equipped with the Orlicz norm are given. Although, criteria for uniform nonsquareness and locally uniform nonsquareness in Orlicz function spaces were known, we can easily deduce them from our main results. Moreover, we give a sufficient condition for an Orlicz–Bochner function space to have the fixed point property.

Approximative compactness and Asplund property in Banach function spaces and in Orlicz–Bochner spaces in particular with application

In this paper it is shown that: (1) If every weak⁎ hyperplane of X⁎ is approximatively compact, then (a) X is an Asplund space; (b) X⁎ has the Radon–Nikodym property. (2) Criteria for approximative compactness of every weakly⁎ hyperplane of Orlicz–Bochner function spaces equipped with the Orlicz norm are given. (3) If X has a Fréchet differentiable norm, then (a) Orlicz–Bochner function spaces LM0(X⁎) have the Radon–Nikodym property if and only if M∈Δ2; (b) Orlicz–Bochner function spaces EN(X) are Asplund spaces if and only if M∈Δ2. (4) We give an important application of approximative compactness to the theory of generalized inverses for operators between Banach spaces and Orlicz–Bochner function spaces.

Formula omitted]-convexity of Orlicz–Bochner function spaces endowed with the Orlicz norm

In this paper, criteria for P -convexity of Orlicz–Bochner function spaces equipped with Orlicz norm are given. Although, criteria for P -convexity of Orlicz function spaces equipped with the Orlicz norm were known, we can easily deduce them from our main results.

Nearly strict convexity in Musielak–Orlicz–Bochner function spaces

Criteria for nearly strict convexity of Musielak–Orlicz–Bochner function spaces equipped with the Luxemburg norm are given. We also prove that, in Musielak–Orlicz–Bochner function spaces generated by strictly convex Banach space, nearly strict convexity and strict convexity are equivalent.

Noncreasy and uniformly noncreasy Orlicz–Bochner function spaces

We point out that uniform noncreasiness is a super-property in Banach spaces. Moreover, we prove that Orlicz–Bochner function spaces are noncreasy if and only if they are rotund or smooth. Finally, we obtain that Orlicz–Bochner function spaces are uniformly noncreasy if and only if they are uniformly rotund or uniformly smooth.

Complex extreme points and complex convexity in Orlicz–Bochner function spaces

It is well known that extreme points which are connected with strict convexity of the whole spaces, are the most basic and important geometric points in geometric theory of Banach spaces. In this paper, criteria for complex extreme points, complex strict convexity and complex uniform convexity in Orlicz–Bochner function spaces are given.

Nonsquareness and Locally Uniform Nonsquareness in Orlicz-Bochner Function Spaces Endowed with Luxemburg Norm

Criteria for nonsquareness and locally uniform nonsquareness of Orlicz-Bochner function spaces equipped with Luxemburg norm are given. We also prove that, in Orlicz-Bochner function spaces generated by locally uniform nonsquare Banach space, nonsquareness and locally uniform nonsquareness are equivalent.

Drop properties and approximative compactness in Orlicz–Bochner function spaces

The representation of dual spaces of E M ( μ , X ) , owing to its extensive application, is given in this paper. Using the representation, we get the sufficient and necessary conditions of L M ( μ , X ) possessing drop property, and extend the result of Hudzik and Wang [H. Hudzik, B. Wang, Approximative compactness in Orlicz spaces, J. Approx. Theory 95 (1998) 82–89]. Simultaneously, under some conditions, the weak ∗ drop property in L M ( μ , X ∗ ) and L M ( μ , X ) ∗ is discussed.

Rotundity of Orlicz-Bochner space with Luxemburg norm

In this paper, by using the generating function Φ and X the criteria of the rotundity for the Orlicz-Bochner function and sequence spaces endowed with the Luxemburg norm were obtained. And the sufficient and necessary conditions were given for the extreme point of Orlicz-Bochner sequence space and the conditions in part for Orlicz-Bochner function space.

A note on the convergence of Orlicz-Bochner spaces with the Luxemburg norm

In this paper, the convergence of the Orlicz-Bochner function spaces with the Luxemburg norm was proved.