A general Fatou Lemma is established for a sequence of Gelfand integrable functions from a vector Loeb space to the dual of a separable Banach space or, with a weaker assumption on the sequence, a Banach lattice. A corollary sharpens previous results in the …nite dimensional setting even for the case of scalar measures. Counterexamples are presented to show that the results obtained here are sh...

We obtain a pointwise description of functions belonging to function spaces with the lattice property. In particular, it is valid for Banach provided that Hardy–Littlewood maximal operator bounded. also study weighted grand Sobolev spaces, defined on arbitrary open sets ? (of finite or infinite measure) in R n , and provide them.

Let X and Y be compact Hausdorff spaces, and E, F be Banach lattices. Let C(X,E) denote the Banach lattice of all continuous E-valued functions on X equipped with the pointwise ordering and the sup norm. We prove that if there exists a Riesz isomorphism Φ : C(X,E) → C(Y, F ) such that Φf is non-vanishing on Y if and only if f is non-vanishing on X, then X is homeomorphic to Y , and E is Riesz i...

We settle several questions regarding the model theory of Nakano spaces left open by the PhD thesis of Poitevin [Poi06]. We start by studying isometric Banach lattice embeddings of Nakano spaces, showing that in dimension two and above such embeddings have a particularly simple and rigid form. We use this to show show that in the Banach lattice language the modular functional is definable and t...

Some classical inequalities are known also in a more general form of Banach lattice norms and/or in continuous forms (i.e., for 'continuous' many functions are involved instead of finite many as in the classical situation). The main aim of this paper is to initiate a more consequent study of classical inequalities in this more general frame. We already here contribute by discussing some results...

We investigate the sufficient condition under which each positive b-weakly compact operator is Dunford-Pettis. We also investigate the necessary condition on which each positive b-weakly compact operator is Dunford-Pettis. Necessary condition on which each positive b-weakly compact operator is weakly compact is also considered. We give the operator that is semi-compact, but it is not bweakly. W...

In this article we discuss lattice convexity and concavity of Calderón-Lozanovskii space Eφ , generated by a quasi-Banach space E and an increasing Orlicz function φ. We give estimations of convexity and concavity indices of Eφ in terms of Matuszewska-Orlicz indices of φ as well as convexity and concavity indices of E. In the case when Eφ is a rearrangement invariant space we also provide some ...

In this paper, we first characterize the continuity of a map in space $ \mathcal{C} = BC(\mathcal{H},\mathbb{R}^m) equipped with compact open topology. Then show that linear lattice and nonlocal dispersal equations generate uniformly continuous semigroups Banach \mathcal{B} supremum norm. Finally, illustrate how to prove nonlinear monotone semiflows respect

We consider analytic coupled map lattices over Z with exponentially decaying interaction. We introduce Banach spaces for the infinite-dimensional system that include measures with analytic, exponentially bounded finite-dimensional marginals. Using residue calculus and ‘cluster expansion’-like techniques we define transfer operators on these Banach spaces. For these we get a unique probability m...

Let X and Y be completely regular spaces and E and F be Hausdorff topological vector spaces. We call a linear map T from a subspace of C(X, E) into C(Y, F ) a Banach-Stone map if it has the form Tf(y) = Sy(f(h(y)) for a family of linear operators Sy : E → F , y ∈ Y , and a function h : Y → X. In this paper, we consider maps having the property: (Z) ∩ki=1 Z(fi) 6= ∅ ⇐⇒ ∩ k i=1Z(Tfi) 6= ∅, where ...