Some remarks on 2-isometry mappings in linear 2-normed spaces

Let R be a normed linear space, K be an arbitrary convex subset of an n-dimensional subspace Φn⊂R. This paper first gives a general charactaerization for a best approximation from K in form of “zero in the convex hull”. Applying it to the uniform approximation by generalized polynomials with restricted ranges, we get further an alternation characterization. Our results ocntains the special cases of interpolatory approximation, positive approximation, copositive approximation, and the classical characterizations in forms of convex hull and alternation in approximation without restriction.

We develop a theory of simultaneous metric projection in a normed linear space $X$ and present various characterizations of simultaneous metric projection onto closed convex sets in terms of the elements of $X^{\ast}$. Also, we characterize the elements of simultaneous metric projection onto closed convex sets in terms of extreme points of the closed unit ball $B_{X^{\ast}}.$ Finally, as an application, we give various characterizations of simultaneous metric projection onto subspaces of the Banach space $C(Q,Y).$

Here we want to present briefly some results, problems and directions of research in the modern theory of best approximation, i.e. in which the methods of functional analysis are applied in a consequent manner. In this theory the functions to be approximated and the approximating functions are regarded as elements of certain normed linear (or, more generally, of certain metric) spaces of functions and best approximation amounts to finding “nearest points”. The advantages and a brief history of this modern point of view have been described in the Introduction to the monograph [82] and we shall not repeat them here; the material which will be presented in the sequel will be convincing enough, we hope, to prove again that the theory of best approximation in normed linear spaces constitutes both a rigorous theoretical foundation for the existing classical and more recent results in various concrete spaces and a powerfull tool for obtaining new results, solving the new problems which appear.

It is known that an important extremal problem in a linear normed space is the classic problemofSteiner, which consists in finding in the set of this space a point (a point of Steiner) to which the sum of the distances from several fixed points of the space would be minimal[1, p.314].In this problem is assumed that all segments of linear normed space are «homogeneous». However, in practice, their lengths often have different «weight»characteristics. We come to the problem of findingin the setof the linear normed space of such a point, that the sum of the weight distance from several fixed points of this space to this pointwould be minimal[2, p. 468;3;4].The generalizedSteiner’s problem in a polynormed space is considered in the article.This problem is obtained as a result of replacing in classic Steiner's problem the sum of the distances from fixed points of linear space to the points of the set of its admissible elements, which are determined by one norm by the sum of the distances from the above-mentioned points with positive weighting coefficients, which are determined by the corre-sponding different norms determined on this linear space.It is clear that the above extremal problems are special cases of the generalized Steiner’s problem in polynormed space.A special case of this problem is also the problem of the best approxi-mation of an element of a linear normalized space by a convex set of this space, which has been studied by many authors.The main results of research for the problem of the best approximation of an element of a linear normed space are summarized, in particular, in the monographs of N.I. Ahiezer [5], V.K. Dzyadyk [6], M.P. Korney-chuk [7], O.I. Stepants [8,9] and others.In this article the conditions of extremalof an allowable element for the generalized problem of steiner in polynormed space, which generalize the re-sults obtained, in particular for the above special cases are established

Characterizations on 2-isometries

In this paper, we use the denition of fuzzy normed spaces given by Bag and Samanta and the behaviors of solutions of the additive functional equation are described. The Hyers-Ulam stability problem of this equation is discussed and theorems concerning the Hyers-Ulam-Rassias stability of the equation are proved on fuzzy normed linear space.

The aim of this paper is to investigate when a linear normed space is an inner product space. Several conditions in a linear normed space are formulated with the help of inequalities. Some of them are from the literature and others are new. We prove that these conditions are equivalent with the fact that the norm is induced by an inner product. One of the new results is the following: in an inner product space, the sum of opposite edges of a tetrahedron are the sides of an acute angled triangle. The converse of this result holds also. More precisely, this property characterizes inner product spaces. Another new result is the following: in a tetrahedron, the sum of squares of opposite edges are the lengths of a triangle. We prove also that this property characterizes inner product spaces. In addition, we give simpler proofs to some theorems already known from the publications of other authors.

This chapter, in fact, starts our study of functional analysis. Here, we consider linear normed spaces, i.e., sets equipped with the structure of a linear space and a norm (an analog of the notion of the length of a vector). These spaces are one of the main objects of investigation in functional analysis. Banach spaces or spaces of type B (complete with respect to convergence in norm) are the most important class of linear normed spaces. These spaces were introduced by Banach and Wiener. Here, we also consider another important class of spaces — Hilbert spaces — whose norm is defined in terms of a scalar product.

The mutational equations of Aubin extend ordinary differential equations to metric spaces (with compact balls). In first-order geometric evolutions, however, the topological boundary need not be continuous in the sense of Painleve–Kuratowski. So this paper suggests a generalization of Aubin’s mutational equations that extends classical notions of dynamical systems and functional analysis beyond the traditional border of vector spaces: Distribution– like solutions are introduced in a set just supplied with a countable family of (possibly non-symmetric) distance functions. Moreover their existence is proved by means of Euler approximations and a form of “weak” sequential compactness (although no continuous linear forms are available beyond topological vector spaces). This general framework is applied to a first-order geometric example, i.e. compact subsets of the Euclidean space evolving according to the nonlocal properties of both the current set and its proximal normal cones. Here neither regularity assumptions about the boundaries nor the inclusion principle are required. In particular, we specify sufficient conditions for the uniqueness of these solutions.

Chapter 5 - Vector Spaces, Hilbert Spaces, and the L2 Space

In this note an analytic representation is given for continuous linear operators from C ( X ) C(X) into a linear normed space Y where C ( X ) C(X) is the space of continuous functions on [0, 1] with values in a linear normed space X.

This chapter develops the beginnings of abstract functional analysis, a subject designed to study properties of functions by treating the functions as the members of a space and formulating the properties as properties of the space. Section 1 defines Banach spaces as complete normed linear spaces and gives a number of examples of these. The space of bounded linear operators from one normed linear space to another is a normed linear space, and it is a Banach space if the range is a Banach space. Sections 2–3 concern Hilbert spaces. These are Banach spaces whose norms are induced by inner products. Section 2 shows that closed vector subspaces of such a space have orthogonal complements, and it shows the role of orthonormal bases for such a space. Section 3 concentrates on bounded linear operators from a Hilbert space to itself and constructs the adjoint of each such operator. Sections 4–6 prove the three main abstract theorems about the norm topology of general normed linear spaces—the Hahn–Banach Theorem, the Uniform Boundedness or Banach–Steinhaus Theorem, and the Interior Mapping Principle. A number of consequences of these theorems are given. The second and third of the theorems require some hypothesis of completeness. The topic of Hilbert and Banach spaces continues in Chapter IV of the companion volume, Advanced Real Analysis.

A.G. Vulih has shown how an essentially unique intrinsic multiplication can be defined in certain types of Riesz spaces (vector lattices) L. In general, the multiplication is not universally defined in L, but L can always be imbedded in a large space L# in which multiplication is universally defined. If ф is a normal integral in L, then ф can be extended to a normal integral on a large space L1(ф) in L#, and L1(ф) may be regarded as an abstract integral space. A very general form of the Radon-Nikodym theorem can be proved in L1(ф), and this can be used to give a relatively simple proof of a theorem of Segal giving a necessary and sufficient condition that the Radon-Nikodym theorem hold in a measure space. In another application, the multiplication is used to give a representation of certain Riesz spaces as rings of operators on a Hilbert space.

Abstract. In this paper, we consider and study a new class of general-ized vector quasi-variational type inequalities and obtain some existencetheorems for both under compact and noncompact assumptions in topo-logical vector spaces without using monotonicity. For the noncompactcase, we use the concept of escaping sequences. 1. IntroductionThe vector variational inequality problem was initiated by Giannessi [11] in nite dimensional Euclidean spaces with applications. Later on many authors[6, 7, 12, 15, 24] generalized vector variational inequalities in abstract spacesin several ways. The vector variational-like inequality is one of the generalizedforms of vector variational inequalities [11]. In 1989, Parida et al [19] stud-ied the existence of solutions for variational-like inequalities in R n space andhave shown a relationship between variational like inequalities problems andconvex programming as well as with complementarity problems. The vectorvariational like inequality and generalized vector variational like inequality arepowerful tools to study non-convex vector optimization problems and convexand nondi erentiable vector optimization problems respectively, see [10, 18, 21].In 1973, Bensoussan and Lions [3] introduced the quasi-variational inequalityproblem (QVIP). Since then, many generalizations of the QVIP have beenappeared in the literature (see, for example, [1, 2]). In the last decade, becauseof applications in the optimization problems, mathematical programming andequilibrium problems, the QVIP has been intensively studied by many authors[5, 8, 16, 20, 22, 23, 25].In this paper, we consider the generalized vector quasivariational type in-equality problem (GVQVTIP) and obtain some existence theorems for solutions

This paper focuses on various textual elements in prose writing (footnotes, titles, and prefaces) such as they determine, and are determined by, the textual spacing of the page. Far from functioning in the way that the typographical, syntactical spaces between words do, or from replicating the pauses punctuating words in speech, such spacing is singularly and irreducibly textual. Functioning like non-phonetic marks such as parentheses, or expressing relations of hierarchy (as between footnote and text), textual spacing’s manifold functions belie their pristine blankness. I look at this textual spacing of the preface, title, and footnote, as its orthodox and deviational modes of functioning come to light in certain of Derrida’s readings. A second phase of analysis focuses on the unconventional spatial organization of certain of Derrida’s own texts—on his juxtaposition of cited texts in a determinate spatial configuration, and his composing of a book comprising two columns facing one another. To make sense of such gestures in the face of the derision they can occasion, I have recourse not, as one might, to French modernist art’s ostentatious engagement and display of its own material conditions, but to the conventional exploitation of textual space as analyzed in the first section devoted to the spacing of footnotes, prefaces, and so on. I conclude Derrida’s gestures to continue the tradition in which prose writing has innovatively availed of textual space to institute such conventional textual components as footnotes, prefaces, and titles—components integral to the apparatus of “the book” that gives material form to or “realizes” the logos. Derrida’s gestures are thus to be judged, not on the basis of the bemusement we might feel at an instance and mode of textual spacing that falls under no familiar convention, but only vis-a-vis their deconstruction of the text as an exhaustible totality of sense, and their visually attesting to the intertextuality interwoven in any text’s composition.