Real Normed Space Research Articles

On the Hyers-Ulam Stability of Linear Mappings

Let E 1 be a real normed vector space and E 2 a real Banach space. S. M. Ulam posed the problem: When does a linear mapping near an approximately linear mapping ƒ: E 1 ↦ E 2 exist? We give a new generalized solution to this problem. An example illustrates when the answer to this question is negative. The behaviour of bounded approximately additive mappings which do not satisfy Hyers-UIam stability is also investigated.

On the Hyers-Ulam Stability of ψ-Additive Mappings

Let E1 be a real normed vector space and E2 a real Banach space. S. M. Ulam posed the problem: When does a linear mapping near an approximately additive mapping ƒ: E1 → E2 exist? We give a new generalized solution to Ulam′s problem for ψ-additive mappings. Some relations with the asymptotic differentiability are also indicated.

On the Mazur-Ulam theorem and the Aleksandrov problem for unit distance preserving mappings

Let X X and Y Y be two real normed vector spaces. A mapping f : X → Y f:X \to Y preserves unit distance in both directions iff for all x , y ∈ X x,y \in X with | | x − y | | = 1 ||x - y|| = 1 it follows that | | f ( x ) − f ( y ) | | = 1 ||f(x) - f(y)|| = 1 and conversely. In this paper we shall study, instead of isometries, mappings satisfying the weaker assumption that they preserve unit distance in both directions. We shall prove that such mappings are not very far from being isometries. This problem was asked by A. D. Aleksandrov. The first classical result that characterizes isometries between normed real vector spaces goes back to S. Mazur and S. Ulam in 1932. We also obtain an extension of the Mazur-Ulam theorem.

The metric injective hulls of normed spaces

Let M denote the category of metric spaces with contractions as morphisms and N denote the category of real normed spaces with linear contractions as morphisms. Given any normed space X over the reals R , let E m ( X), E n ( X) denote the injective hulls of X in the categories M and N respectively. It was shown by Isbell that E m ( X) “coinsides” with E n ( X) but his proof had a gap. In this paper we shall fix that but also provide two other direct ways of getting at the Banach algebra structure of E n ( X).

On the Clarke subdifferential of the distance function of a closed set

Let C be a nonempty closed subset of the real normed linear space X. In this paper we determine the extent to which formulas for the Clarke subdifferential of the distance for C, d c(x) := inf y∈c ‖x − y‖ , which are valid when C is convex, remain valid when C is not convex. The assumption of subdifferential regularity for d c plays an important role. When x ∉ C, the most precise results also require the nonemptiness of the set P C(x) := { x ̄ ϵ C : ∥x − x ̄ ∥ = d C(x)} . As an interesting side result, the equivalence between the strict differentiability and the regularity of d C at x is established when x ∉ C, P C ( x) ≠ Ø, and the norm on X is smooth.

What are the linear isometries of 1p/n?

We use elementary methods to show that, if p= 2 then the linear isometries of the real normed space 1p/n are given by the nxnsigned permutation matrices. This result, though known to Banach as far back as the 1930s, does not appear to be well known to the general mathematical audience. One reason for this obscurity might be that the given description is usually obtained as an application of rather ‘difficult’ theorems about isometries of a general class of normed spaces. The approach we take is straightforward. When p= 1 or p=8 we consider the action of any isometry on the extreme points of the unit ball (which are finite in number) to obtain the result. For 1 <p< 8 and =2 the basic technique is to differentiate certain equations involving the norm with respect to a parameter.

Korovkin inequalities for stochastic processes

The rate of convergence of a sequence of positive linear stochastic operators { T j } j ∈ n to the unit operator I on spaces of stochastic processes X is studied. This is mainly done for stochastic processes that are smooth over a compact and convex index set Q ⊂ R k, k ⩾ 1. Nearly best upper bounds are given for ¦E(T jX)(x 0) − (EX)(x 0)¦, x 0 ϵ Q , where E is the expectation operator and T j are E-commutative. These lead to strong and elegant inequalities many times sharp, which involve the first modulus of continuity of EX α, where X α is a (partial) derivative of X. The case of Q being a compact convex subset of a real normed vector space is also met and there the upper bound is the best possible.

On a valuation field invented by A. Robinson and certain structures connected with it

We clarify the structure of the non-archimedean valuation field ρ R which was introduced by A. Robinson, and of theρ-non-archimedean hulls of Banach algebras and Lie groups. (For Banach spaces this construction is due to W. A. J. Luxemburg.) In particular, we show that any two infinite-dimensional real normed spaces have a pair of isometrically isomorphicρ-non-archimedean hulls.

Characterizations of proximinal, semichebychevian and chebychevian subspaces in real normed spaces

Characterizations of proximinal, semichebychevian and chebychevian subspaces in real normed spaces

Finite extremal characterization of strong uniqueness in normed spaces

A strongly unique best approximation m in a finite-dimensional subspace M of a real normed linear space X to an element x ϵ X/ M is characterized by means of a finite number of extremal points of the closed unit ball in the dual space X ∗ . This result is applied to weak Chebyshev subspaces in C( T).

Chebyshev coefficients for $L^1$-preduals and for spaces with the extension property

We apply the Chebyshev coefficients ?f and ?b, recently introduced by the authors, to obtain some results related to certain geometric properties of Banach spaces. We prove that a real normed space E is an L1-predual if and only if ?f(E) = 1/2, and that if a (real or complex) normed space E is a P1 space, then ?b(E) equals ?b(K), where K is the ground field of E.

Fundamental theorems of nondominated solutions associated with cones in normed linear spaces

Essential properties of multiobjective programming in real normed linear spaces are studied. Necessary and sufficient conditions for nondominated solutions under regularity and Fréchet differentiability assumptions are developed.

Quasi-additive functions

Let 0 ⩽e < 1 and letX, Y be real normed spaces. In this paper we consider the following functional inequality:∥f(x + y) − f(x) − f(y)∥ ⩽ e min{∥f(x + y)∥, ∥f(x) + f(y)∥} forx, y ∈ R, wheref: X → Y. Mainly continuous solutions are investigated. In the case whereY = R some necessary and some sufficient conditions for this inequality are given.

On a Theorem of Arrow, Barankin, and Blackwell

A well-known theorem of Arrow, Barankin, and Blackwell states that if ${\bf R}^n $ is equipped with the natural ordering, then for every compact convex subset S of ${\bf R}^n $ the set of properly minimal elements of S is dense in the set of minimal elements of S. In this note a result of Jahn is used to show a generalization of the density theorem of Arrow, Barankin, and Blackwell. It will be shown that this theorem holds in a real normed space that is partially ordered by a convex cone with a closed bounded base.

Strong unicity in normed linear spaces

We consider best approximations in a real or complex normed linear space by elements of a finite dimensional subspace. It is the purpose of this paper to characterize, when a best approximation to a given element is strongly unique.

Infima of Convex Functions

Let $\Gamma (X)$ be the lower semicontinuous, proper, convex functions on a real normed linear space $X$. We produce a simple description of what is, essentially, the weakest topology on $\Gamma (X)$ such that the value functional $f \to \inf f$ is continuous on $\Gamma (X)$. When $X$ is reflexive, convergence of a sequence in this topology is equivalent to Mosco plus pointwise convergence of the corresponding sequence of conjugate convex functions.

Strong laws of large numbers for weighted sums of random elements in normed linear spaces

Consider a sequence of independent random elements {Vn, n > in a real separable normed linear space (assumed to be a Banach space in most of the results), and sequences of con-

The rate of weak convergence of convex type positive finite measures

Let μ be a positive finite measure of mass m of the non-empty convex subset M of the real normed vector space ( V, ∥ · ∥). For a fixed x 0 ϵ M and τ( x) = ∥ x − x 0∥, let the probability measure ϱ = m −1 μ ∘ τ −1. Assume that the corresponding ϱ distribution function fulfills certain convexity conditions. By the use of convex moment methods best upper bounds of ¦∫ M f dμ − f(x 0)¦ are obtained for f an integrable real valued function on M and a given power moment of μ. These lead to sharp inequalities, i.e., attainable inequalities involving the first modulus of continuity of f. The established estimates improve the corresponding ones in the literature. These have wide applications to concave positive linear operators typically arising from well-known probabilistic distributions.

A Generalization of a Theorem of Arrow, Barankin, and Blackwell

This paper presents a generalization of a known density theorem of Arrow, Barankin, and Blackwell (“Admissible points of convex sets,” in Contributions to the Theory of Games, H. W. Kuhn and A. W. Tucker, eds., Princeton University Press, Princeton, NJ, 1953). This result is shown to hold even in a real normed space partially ordered by a Bishop-Phelps cone.

Norms in product spaces which preserve approximation properties

SynopsisLet E1 and E2 be real normed linear spaces such that the dimension of any of them is at least 2. We prove that the norms in E1 × E2 which verify a simple property of monotonicity with regard to the initial norms in E1 and E2 are the only norms in E1 × E2 which preserve best linear approximations, in the sense that ifyk ∊ Lk is best approximation to xk from the linear subspace Lk, (k = 1,2), then (y1, y2) is best approximation to (x1, x2) from L1 × L2.