On sharp triangle inequalities in banach spaces pdf

Banach spaces these notes provide an introduction to banach spaces, which are complete normed vector spaces. The concept of sharp efficient solution for vector equilibrium problems on banach spaces is proposed. F pq f qr involves the binary operation of convolution on the space of distribution functions. On the volume of unit balls of finitedimensional lorentz. In 7, kato, saito and tamura presente d the following sharp triangle inequality in a banach space as follows. Moreover we consider equality attainedness for sharp triangle inequality and its reverse inequality in strictly convex banach spaces. For the purposes of these notes, all vector spaces are assumed to be over the real numbers. The triangle inequality is one of the most fundamental inequalities in analysis and has been studied by several authors.

Several standard results true in greater generality have simpler proofs for banach spaces. They give bounds for the l pnorms of the sum and difference of two measurable functions in l p in terms of the l pnorms of those functions individually statement of the inequalities. We shall present a couple of norm inequalities which will much improve the sharp triangle inequality with n elements and its reverse inequality in a banach. The existence of a solution for a class of variational inequalities in banach spaces is presented. Herewe consider another aspect of the classical triangle inequality x y. Complete metric spaces notions such as convergent sequence and cauchy sequence make sense for any metric space. Godefroykalton 2003 let xand ybe separable banach spaces and suppose that f. Apr 28, 2010 sharp triangle inequality and its reverse in banach spaces were recently showed by mitani et al. Inequalities for the normalized jensen functional with. Particular instances in the case of normed spaces and for complex and real ntuples are given. Riesz lemma is an elementary result often an adequate substitute in banach spaces for the lack of sharper. A banach space b is a real vector space that is equipped withanorm that satis. Existence theorems for elliptic quasivariational inequalities in banach spaces risei kano. In this paper, we study the properties of the metric projection operator the nearest point projection operator and.

The triangle inequality is one of the most important and fundamental inequalities in analysis. It is based only on the standard fractional gagliardonirenberg inequality and a fractional chain rule. This process is experimental and the keywords may be updated as the learning algorithm improves. We also mention a very general setting for nonlinear approximation in banach spaces introduced by temlyakov et al. Generalized dunklwilliams inequality in 2inner product.

In mathematics, more specifically in functional analysis, a banach space pronounced is a complete normed vector space. Let be a real banach space with norm and dual space. For all nonzero elements x, y in a normed linear space x with. On sharp triangle inequalities in banach spaces, j. On the triangle inequality in quasibanach spaces rims, kyoto. To extend the result to the full scale of parameters, we. Normed and banach spaces in this chapter we introduce the basic setting of functional analysis, in the form of normed spaces and bounded linear operators. Complementary triangle inequality in hilbert space. We shall present a couple of norm inequalities which will much improve the sharp triangle inequality with n elements and its reverse inequality in a banach space shown recently by the last three. Banach spaces rather fragmented, maybe you could say it is underdeveloped, but one can argue that linear approximations are often used for considering nonlinear problems.

Dec 15, 2007 we shall present a couple of norm inequalities which will much improve the sharp triangle inequality with n elements and its reverse inequality in a banach space shown recently by the last three authors. Reverses of the triangle inequality in banach spaces sever s. The first sharp gyrotriangle inequality in mobius gyrovector. Triangle inequality is regarded as the most prominent and important inequality in the theory of normed spaces and therefore it has been a major topic of interest treated by numerous mathematicians. Triangle inequality an overview sciencedirect topics.

Preliminaries in this section, we will gather together some of the necessary preliminary information on banach spaces and metric spaces which we will use later in the paper. A vector space xtogether with a norm k k is called a normed linear space, a normed vector space, or simply a normed space. Banach and hilbert spaces in what follows k will denote r of c. A banach space x is said to have azuma type p, if there exists a constant k 0 depending on x so that for every martingale f fj. Many nonlinear problems in functional analysis can be reduced to the search of fixed points of nonlinear operators. We extend coulombsobolev inequalities associated to the space es. On sharp triangle inequalities in banach spaces ii core. In, dunkl and williams proved that in inequality 1.

In this paper, we show the triangle inequality and its reverse inequality in quasibanach spaces. On sharp triangle inequalities in banach spaces ii. Introduction in recent years, the triangle inequality and its reverse inequality have been treated in 15. Introduction jensens inequality is one of the most fundamental and extensively used inequalities. In recent years, the triangle inequality and its reverse inequality have been treated in 15 see also 6, 7. All vector spaces are assumed to be over the eld f.

After that, we shall prove some smoothing results for lpfunctions. Banach space hardy space singular integral operator convex banach space sharp inequality these keywords were added by machine and not by the authors. Pdf sharp triangle inequalities in quasinormed spaces. Functional analysis is a wonderful blend of analysis and algebra, of. Sharp jackson and bernstein inequalities for nterm. Funtional analysis lecture notes for 18 mit mathematics. We shall present a couple of norm inequalities which will much improve the sharp triangle inequality with n elements and its reverse inequality in a banach space shown recently by the last three authors. In this paper, we present equality attainedness for these inequalities in strictly convex banach spaces. In a recent paper l, wilf has given an extension of the arithmeticgeometric mean inequality to the case of complex numbers. In mathematics, a normed vector space is a vector space on which a norm is defined. Completeness for a normed vector space is a purely topological property.

Abstract sharp triangle inequality and its reverse in banach spaces were recently showed by mitani et al. Bounds for the normalised jensen functional bulletin of. Request pdf sharp triangle inequality and its reverse in banach spaces we shall present a sharp triangle inequality and its reverse inequality with n. On the sharp triangle inequalities in quasibanach spaces.

On sharp triangle inequalities in banach spaces ii by kenichi mitani and kichisuke saito get pdf 274 kb. On sharp triangle inequalities in banach spaces ii springerlink. New inequalities for the general case of convex functions defined on linear spaces which improve the famous jensens inequality are established. Furthermore, a quasinormed space is said to be a p. Many authors have been treating its generalizations and reverse inequalities. Do cauchys mean value theorem or lhopitals rule have natural generalizations in the context of maps between banach spaces. Here one can again interpret geometrically the hypothesis as requiring certain vectors to lie within a cone.

Bregman projection operator with applications to variational. In this paper,we shall present the sharp triangle inequality and its reverse inequality for an arbitrary number of finitely many nonzero elements of a quasi banach space, which generalize. We make no assumptions about the meaning of the symbol x, i. In mathematics, clarksons inequalities, named after james a. A norm is a realvalued function defined on the vector space that has the following properties. If the inline pdf is not rendering correctly, you can download the pdf file here. On sharp triangle inequalities in banach spaces ii journal. Generalized system of variational inequalities in banach spaces. In 7, kato, saito and tamura presente d the following sharp triangle inequalityin a banach space as follows. In this paper we prove the reverse of triangle inequality via selbergs inequalities in the framework of hilbert cmodules. Carlen, frank, ivanisvili and lieb 2018 recently obtained. Research article on sharp triangle inequalities in banach. In terms of the primal gap function associated to the mixed variational inequality, we give several characterizations of the weak sharpness.

Abstractly, banach spaces are less convenient than hilbert spaces, but still su ciently simple so many important properties hold. Refinements of shannons inequality and the positivity of kullbackleibler divergence are obtained. However, the above inequality, as well as its generalizations are also very important in the theory of quasinormed spaces. A short course on non linear geometry of banach spaces 3 we nish this very short section by mentioning an important recent result by g. Pdf on sharp triangle inequalities in banach spaces ii. Sharp triangle inequality and its reverse in banach spaces. Throughout, f will denote either the real line r or the complex plane c.

Pdf on the triangle inequality in quasibanach spaces. Generalized system of variational inequalities in banach spaces abdellah bnouhachem1,2. Introduction jensens inequality is one of the most fundamental and extensively used inequalities in analysis and other fields of mathematics. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of length in the real world. Feb 01, 2016 on sharp triangle inequalities in banach spaces, j. Sharp triangle inequality and its reverse in banach spaces request. In this paper,we shall present the sharp triangle inequality and its reverse inequality for an arbitrary number of finitely many nonzero elements of a quasi banach space.

In this paper, we present equality attainedness for these inequalities in strictly convex. The main result that we will prove in this paper shows that the decomposition. Weak sharp solutions for variational inequalities in banach. Of course, this means that we have to start caring about.

There are many fundamental inequalities in mathematics and one of them is the famous triangle inequality. A normed space is a pair x,kk, where xis a linear space. Sharp efficiency for vector equilibrium problems on banach spaces. A banach space over k is a normed kvector space x,k. A metric space is a set xtogether with a metric don it, and we will use the notation x. Namely, we characterize equality attainedness of inequalities 1. We shall discuss some inequlaities appearing in the geometry of banach spaces. In this paper, using the approximate duality mapping, we introduce the definition of weak sharpness of the solution set to a mixed variational inequality in banach spaces. On sharp triangle inequalities in banach spaces ii article pdf available in journal of inequalities and applications 20105, art. An industry standard proof of lhopitals rule depends on cauchys mean value theorem and i am not sure whether this admits a banach space analogue either. Applying the same reasoning using the integral version of jensens inequality gives p q z x fpd 1p x fqd 1q for any l1 function f. A study on copson operator and its associated sequence space. Hilbert spaces with respect to the inner product hf,gi z fg the integral to be taken over the appropriate domain.

A normed linear space x is called a banach space if it is complete, i. On sharp triangle inequalities in banach spaces sciencedirect. Recent reverses for the discrete generalised triangle inequality and its continuous version for vectorvalued integrals in banach spaces are surveyed. The triangle inequality in lorentz spaces 5557 as in the case of the decomposition norm, p,sis a norm, equivalent to with f p,s f,if1. Envelopes between l2 and lp spaces paata ivanisvili and connor mooney abstract.

Y is an into isometry, then xis linearly isometric to a subspace of y. By using a mean operator we shall present some sharp mean triangle inequalities in a banach space which generalize the sharp triangle inequality with n elements and its reverse one shown recently by the last three authors in 7. In this paper, we introduce the notion of a weak sharp set of solutions to a variational inequality problem vip in a reflexive, strictly convex and smooth banach space, and present its several equivalent conditions. Reverse of the triangle inequality in hilbert cmodules. Norms a norm is a function that measures the lengths of vectors in a vector space. This follows from the next lemma, which is the cauchyschwarz inequality in this setting. In the course of doing this we shall present a new two element triangle inequality with parameter and its reverse. Thus, a banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a cauchy sequence of vectors always converges to a well defined limit that is within the space. Moreover, the fermat rules for local efficient solutions of vector equilibrium problems are extended to the sharp efficient solutions by means of the clarke generalized differentiation and the normal cone. In lectures i proceed to the next chapter, on lebesgue. By using a mean operator, we present some sharp mean triangle inequalities in a banach space which generalize the sharp triangle inequality with n elements and its reverse shown recently by the. Reverses of the triangle inequality in banach spaces. Another approach to characterizations of generalized triangle. Sharp triangle inequality and its reverse in banach spaces were recently showed by mitani et al.

On sharp triangle inequalities in banach spaces request pdf. Banach, spaces and the process of completion of a normed space to a banach space. Pdf in this paper, we show the triangle inequality and its reverse inequality in quasibanach spaces. A hilbert space is thus a banach space whose norm is associated with an inner product. The metric projection and its applications to solving variational inequalities in banach spaces jinlu li department of mathematical sciences shawnee state university portsmouth, ohio 45662, usa email. Inequalities, normed space, norm inequality, triangle inequality, reversed triangle inequality, angular distance, fischermuszely equality. A survey of some recent inequalities for the norm and numerical radius of operators in hilbert spaces dragomir, sever s. Tamura, sharp triangle inequality and its reverse in banach spaces, math. We also investigate the topological properties, inclusions, and dual spaces of these new banach spaces as well as compute the norm of copson operators on the wellknown matrix domains. Serstnev put these two inequalities on an equal footing by lifting the tnorm t to a binary operation on distribution functions.