This paper deals with quasi variational inequality problems qvis in a generic banach space setting. A 0,a1 be a quasi banach couple, let 0 quasi completeness april 24, 2014 is a strict colimit of the subspaces co b rn ff2corn. Quasi compactness in a quasi banach space for the sequence space lp, p banach space attain their supremum on the unit sphere then the space is reflexive. Quasigreedy bases in the spaces 0 quasi banach space yextends to a compact operator t. Moreover, we establish relationships of these moduli and. Abstract we generalize the classical coorbit space theory developed by feichtinger and gro. Pdf quasicompactness in a quasibanach space for the sequence space has been introduced based on the important extension of milmans. The smallest possible is called the modulus of concavity of. A complete quasinormed vector space is called a quasi banach space. Recall that a quasi banach space x is said to be a k space if whenever z is a quasi banach space with a subspace l of dimension one such that z l is isomorphic to x. We provide a theoretical framework for the analysis of such problems which. Throughout the paper, whenever an aconvex space q, ii 9 if appears, we will suppose that ii ii is at the same time rnorm and plurisubharmonic, thus we will have c 1 in 1. Let x and b be quasi banach spaces such that x is a subspace of b.
Interpolation of multilinear operators acting between quasi. Quasi banach spaces of almost universal disposition. On generalized moduli of quasibanach space hindawi. By contrast the study of quasi banach space has lagged far behind, even though the first research papers in. In this article, we introduce and study an iterative algorithm which is a combination of general iterative method with strongly accretive operator and generalized viscosity explicit methods gvem for finding fixed points of quasi nonexpansive mappings in banach spaces.
Approximate quartic and quadratic mappings in quasibanach spaces. For 1 quasi seminorm, can be reformulated as the geometric assertion that two convex subsets of a normed vector space can be separated by a hyperplane whenever one of them has nonempty interior. Lately many researchers have been interested about diverse issues related to quasibanach spaces. Our starting point is an integrable representation. However there is a metric that generates the same topology. For the purposes of these notes, all vector spaces are assumed to be over the real numbers. In the present paper we will present the triangle inequality in quasi normed spaces. The results of this paper extend a host of previously known results for metric space in a quasi2banach space. Following 14, definition 2, let us say that a quasi banach space u is of almost universal disposition for the class c if, for every. By proposition 8, is uniformly nonsquare if and only if, where, and by proposition 9, if and only if. X y, where y belongs to c and x is a subspace of u, there is an. Wiener amalgam spaces with respect to quasibanach spaces. The extension to the quasi banach case is justified by the importance of some quasi banach spaces, notably h,, p space and it is actually complete, so it is a banach space.
A quasi norm is a realvalued function on x satisfying the following. Throughout, f will denote either the real line r or the complex plane c. Dynamics and chaotic of polynomials on quasi banach spaces. Banach or quasi banach sequence spaces formed by sequences with special properties. Researcharticle on generalized moduli of quasi banach space youngchelkwun,1 hussainminhajuddinahmadqadri,2 waqasnazeer,3 absarulhaq,4 andshinminkang 5,6 dpartmentofmathematics,dngauniversity,busan, republicofkorea. In partic ular, when the given quasibanach function space is the lebesgue space lprn.
Of course, such a space b exists if and only if a1 contains a reflexive subspace. Pdf isometric additive mappings in generalized quasibanach. On quasicomplemented subspaces of banach spaces, with an. A vector space with an associated quasinorm is called a quasinormed vector space. Lipschitz structure of quasibanach spaces springerlink. Dynamics and chaotic of polynomials on quasi banach spaces naseif jasim aljawari, arwa nazar mustafa department of mathematics, college of science, university of almustansiriyah, baghdad,iraq. Ghorbani department of mathematics yazd university yazd 89195741, iran ghorbani. The next theorem follows from the open mapping theorem see comments previous to 20, theorem 2. For 0 quasi normed space and it is actually complete, so it is a quasi banach space. The first type of ldt in the spirit of stein 1970, characterizes the convergence of quasinorm averages of, where. Pdf quasibanach space for the sequence space lp where 0 quasi compactness in a quasi banach space for the sequence space has been introduced based on the important extension of milmans reverse brunnminkowiski inequality by bastero et al. However the result can be extended to nonseparable quasi reflexive banach space. We achieve these mapping properties by using an interpolation functor that can explicitly generate a given rearrangementinvariant quasi banach function space via lebesgue spaces. We prove that, for a large class of ba nach or quasibanach spaces e of xvalued sequences, the.
We show that the lipschitz structure of a separable quasi banach space does not determine, in general, its linear structure. For more information about quasi banach spaces, the readers can refer to 1. Let be the characteristic of generalized modulus of convexity of a quasi banach space. Banach spaces these notes provide an introduction to banach spaces, which are complete normed vector spaces. Quasibanach spaces of almost universal disposition. Let e i be quasibanach spaces with quasibanach constants c i, i 0, 1, and let m be the quasibanach constant of c then, the interpolation space e 0, e 1. Using the notion of the arenseells p space over a metric space for 0 quasi banach spaces which are lipschitz isomorphic but not linearly isomorphic. Coupled fixed point results on quasibanach spaces with. Spaceability in banach and quasibanach spaces of vector. We remark that b has this property if and only if there exists a bounded linear map from b onto a separable banach space. Note that every a banach space is a quasi banach space with k 1 and every a quasi banach space is a bmetric space with dx. Characterization of partiallygreedy bases in quasibanach spaces in terms of quasigreediness and different conservativelike properties. Young chel kwun, 1 hussain minhaj uddin ahmad qadri, 2 waqas nazeer, 3. Approximate cauchy functional inequality in quasibanach.
These spaces arise in a natural way as a generalization of. So in this paper we extend the classical coorbit space theory to quasibanach spaces. Triangle inequality in quasi banach spaces cong wu and yongjin li vol. Researcharticle on generalized moduli of quasibanach space. Quasi banach space hp space when \p 1\ were also used significantly in alexandrovs solution of the inner function problem in 1982. It seems to be an open question if every banach space b has a quasi complemented separable subspace. Quasibanach spaces are an important class of metrizable topological vector spaces often, not locally convex, 70, 83, 87, 88, 105, 5. On the one hand, this means that perhaps quasi banach rather than banach spaces are the most natural habitat for twisted sums. Let x be a quasi banach space and suppose 0 quasi banach spaces l. Whether this result is true for a general banach space is not known. Characterization of partiallygreedy bases in quasi banach spaces in terms of quasigreediness and different conservativelike properties. Complex interpolation of quasi banach spaces with an aconvex containing space 185 equivalent to 1 i which is plurisubharmonic. Complete metric spaces notions such as convergent sequence and cauchy sequence make sense for any metric space.
A quasi banach space is a complete quasinormed space. In the proof, a fixed point theorem due to dung and hang, which is an extension of a fixed point theorem in banach spaces, plays a main role. Pdf quasibanach space for the sequence space lp where 0 banach space attain their supremum on the unit sphere then the space is reflexive. A complete quasinormed vector space is called a quasibanach space. F is a complex vector space endowed with a quasi normkkf with respect to which this is complete i. Box 80203, jeddah, 21589, saudi arabia 1 young researchers and elite club, rasht branch, islamic azad university, rasht, iran the aim of this. We recall some basic facts concerning quasibanach spaces and some pre liminary results. A quasi banach space is uniformly nonsquare if and only if.
In particular, we have the mapping properties of the laplace transform, the hankel transforms, the kontorovichlebedev transform and some oscillatory integral operators. Banach spaces j muscat 20051223 a revised and expanded version of these notes are now published by springer. In this case, a quasi banach space is called a banach space. Wieneramalgamspaceswithrespectto quasibanachspaces arxiv. Construction of p banach spaces of almost universal disposition let c be a class of quasi banach spaces. V called addition and scalar multiplication, respectively, that obey the following axioms. A quasi norm is called a pnorm 0 quasi banach space is called a p banach space. The chapter also discusses re exive banach spaces and includes an exposition of the james space. On the stability of a parametric additive functional. We continue with the study of greedytype bases in quasibanach spaces started in 3. Given a cpartiallygreedy basis in a quasibanach space. The subject of chapter 3 are the weak topology on a banach. Fixed point theory and applications coupled fixed point results on quasi banach spaces with application to a system of integral equations nawab hussain 0 peyman salimi 1 saleh almezel 0 0 department of mathematics, king abdulaziz university, p.
This associated family of quasi banach function spaces xp, which is intimately connected to the base space x, is produced via a procedure akin to that which produces the lebesgue l pspaces from l 1 or more generally, produces the pconvexification of banach lattices of functions, 9, 99, pp. Banach space attain their supremum on the unit sphere then the space is reflexive. Spaceability in banach and quasibanach sequence spaces core. A great number of such moduli have been defined and introduced since then. In this work, we extend and improve cat0 spaces to bcat0 spaces by using the concept of bmetric spaces. Hilbert, banach, fr echet, and lfspaces, as well as their weakstar duals, and other spaces of mappings such as the strong and weak operator topologies on mappings between hilbert spaces, are quasi complete.
Weak and quasi approximation properties in banach spaces. It is also important to note that quasi normed spaces also obey the kahanekhintchine inequality. The theory of the geometry of banach space has evolved very rapidly over the past fifty years. By khintchines inequality the spaces r and c are both of type and cotype 2. Giovanni leoni, spring 2015 monday, january 12, 2015. A modified generalized viscosity explicit methods for.
This associated family of quasibanach function spaces xp, which is intimately connected to the base space x, is produced via a procedure akin to that which produces the lebesgue l pspaces from l 1 or more generally, produces the pconvexification of banach lattices of functions, 9, 99, pp. Functional inequalities in generalized quasibanach spaces. Bythe aokirolewicz theorem see 33, each quasi norm is. It seems that wiener amalgams with respect to quasi banach spaces have not yet been considered in full generality, except for a few results for wiener amalgams on rd in 10. On the triangle inequality in quasibanach spaces emis. Existence theorems for elliptic quasi variational inequalities in banach spaces risei kano. All vector spaces are assumed to be over the eld f. Introduction let us first recall some basic facts concerning quasi banach spaces and some preliminary results. Completeness and quasi completeness april 24, 2014 is a strict colimit of the subspaces co b rn ff2corn. We show that the lipschitz structure of a separable quasibanach space does not determine, in general, its linear structure. Approximate cauchy functional inequality in quasi banach spaces. Construction of pbanach spaces of almost universal disposition let c be a class of quasibanach spaces. Generalization of the pecaricrajic inequality in a quasi. In this paper, we study partiallygreedy bases focusing our attention in two main results.
On the other hand, an inspection of an earlier construciton of en o, lindenstrauss and pisier 1 led kalton 3 to a crucial realization that. We remark that there are some interesting papers concerning the stability of functional inequalities. When p 1 and q6 1, there cannot be a norm that is equivalent to our quasi norm. Given a cpartiallygreedy basis in a quasi banach space. Complex interpolation of quasibanach spaces with an aconvex. Abstract in the present paper, the concepts of a quasi metric space, quasi banach space. So in this paper we extend the classical coorbit space theory to quasi banach spaces. Pdf a note on partiallygreedy bases in quasibanach. In this paper we prove a fixed point theorem for mappings in quasi2banach space via an implicit relation. For these spaces, new measures of noncompactness are also defined, related to the hausdorff measure of noncompactness. A banach space x is isomorphic to a hilbert space if and only if it.
Given 0 quasi banach sequence spaces are thus characterized as a particular case of s. A standard example was the space of bounded linear operators on a banach space. So this paper deals with basic properties of wiener amalgams wb,y with a quasibanach space y as global component and one of the spaces b l1,l. Following 14, definition 2, let us say that a quasibanach space u is of almost universal disposition for the class c if, for every. In this article we discuss moduli and constants of quasi banach space and give some important properties of these moduli and constants. Theory and augmented lagrangian methods christian kanzow ydaniel steck december 3, 2018 abstract. As a result, we get some more general inequalities. Pdf a note on partiallygreedy bases in quasibanach spaces. Interpolation of bilinear operators between quasibanach spaces. We continue with the study of greedytype bases in quasi banach spaces started in 3. Quasi compactness in a quasi banach space for the sequence space lp, p quasibanach spaces s. So this paper deals with basic properties of wiener amalgams wb,y with a quasi banach space y as global component and. Quasi completeness for fr echet spaces is ordinary metric space completeness.
We also show its hyperstability on a restricted domain. Pdf the aim of this paper is to introduce the concept of a quasibanach space for the sequence space, p 0 p 1. An important breakthrough was the work of roberts in and, who showed that the kreinmilman theorem fails in general quasi banach space by developing powerful new techniques. Isometric additive mappings in generalized quasi banach spaces. On moduli and constants of quasibanach spaces pisrt. The study of banach algebras began in the twentieth century and originated from the observation that some banach spaces show interesting properties when they can be supplied with an extra multiplication operation. In the present paper, we shall give an extension of the well known pecaricrajic inequality in a quasi banach space, we establish the generalized inequality for an arbitrary number of finitely many nonzero elements of a quasi banach space, and obtain the corresponding upper and lower bounds. The second type of ldt establishes conditions for pointwise convergence of the best or extended best constant approximants of. Interpolation of multilinear operators acting between quasi banach spaces l.
Interpolation of bilinear operators between quasibanach. Strict convexity of banach spaces was first introduced in 1936 by. Generalized quasibanach sequence spaces and measures of. Approximate quartic and quadratic mappings in quasibanach. For a quasibanach space, the modulus of smoothness is a function. The pair is called a quasinormed space if is a quasinorm on. The following proposition is the bounded version of proposition 2.