General characterization of best approximation and its application to uniform approximation with restricted ranges

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 $C(Q,Y).$

A characterization of best approximations with restricted ranges

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.

‎In this paper‎, ‎we consider the notion of linear 2-normed space and‎ ‎investigate its unique completion up to 2-isometry as new sense‎. ‎Also a‎ ‎Mazur-Ulam type theorem in linear 2-normed space will appear as a‎ ‎second part of the paper based on certain conditions‎.

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

This paper gives a general characterization theorem of a best uniform approximation of generalized polynomial having multiple restricted ranges of its derivatives. This theorem is widely applicable. The results on characterization in many standard approximations, such as approximation with Hermite-Birkhoff interpolatory side conditions, multiple comonotone approximation, and approximation by algebraic polynomials having bounded coefficients, etc., are special cases of our results.

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.

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

This article studies the strong conical hull intersection property (strong CHIP) for a general (possibly infinite) system of convex sets in normed linear spaces. The first part of the article presents necessary and sufficient conditions for strong CHIP in terms of tangent cones and feasible direction cones in the primal space, and reveals some differences between strong CHIP for finite and infinite systems. The second part of the paper proves that strong CHIP has the segment extension property, i.e. if strong CHIP is satisfied at some ‘base points’ then it can be extended from the base points along line segments to the entire set. Thus, the verification effort of strong CHIP can be greatly reduced in many cases. For example, if the intersection set is a translated cone, then strong CHIP at the vertex ensures strong CHIP for the entire cone. In the special case of Euclidean space, strong CHIP at the extreme points of the intersection set ensures strong CHIP for the entire intersection.

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.

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.

We completely characterize smoothness of bounded linear operators between infinite dimensional real normed linear spaces, probably for the very first time, by applying the concepts of Birkhoff-James orthogonality and semi-inner-products in normed linear spaces. In this context, the key aspect of our study is to consider norming sequences for a bounded linear operator, instead of norm attaining elements. We also obtain a complete characterization of smoothness of bounded linear operators between real normed linear spaces, when the corresponding norm attainment set non-empty. This illustrates the importance of the norm attainment set in the study of smoothness of bounded linear operators. Finally, we prove that Gâteaux differentiability and Fréchet differentiability are equivalent for compact operators in the space of bounded linear operators between a reflexive Kadets-Klee Banach space and a Fréchet differentiable normed linear space, endowed with the usual operator norm.

The notion of weak compactness plays an important role in normed linear space theory. However in many instances norm completeness of the space is an important assumption for obtaining relevant theorems. This note generalises the concept of weak compactness for subsets of normed linear spaces and obtains generalisations of known theorems including that of Eberlein [2]. We use the methods and techniques of “non-standard analysis” in the proofs; several useful non-standard results for normed linear spaces are given.

A linear metric space (X, d) is called a convex linear metric space if for all x, y in X, it also satisfies $$d(\lambda x+(1-\lambda )y,0)\le \lambda d(x,0)+(1-\lambda )d(y,0)$$ whenever $$0\le \lambda \le 1$$ . Such spaces, known to be more general than normed linear spaces, are examples of convex metric spaces extensively discussed in the literature. In this article, we show that convex linear metric spaces are infact normable.