Real Normed Vector Space Research Articles

Isomorphism between Spaces of Multilinear Maps and Nested Compositions over Real Normed Vector Spaces

Summary This paper formalizes in Mizar [1], [2], that the isometric isomorphisms between spaces formed by an (n + 1)-dimensional multilinear map and an n-fold composition of linear maps on real normed spaces. This result is used to describe the space of nth-order derivatives of the Frechet derivative as a multilinear space. In Section 1, we discuss the spaces of 1-dimensional multilinear maps and 0-fold compositions as a preparation, and in Section 2, we extend the discussion to the spaces of (n + 1)-dimensional multilinear map and an n-fold compositions. We referred to [4], [11], [8], [9] in this formalization.

ON UNIFORM CONTRACTIONS OF BALLS IN MINKOWSKI SPACES

Let $N$ balls of the same radius be given in a $d$-dimensional real normed vector space, i.e., in a Minkowski $d$-space. Then apply a uniform contraction to the centers of the $N$ balls without changing the common radius. Here a uniform contraction is a contraction where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers. The main results of this paper state that a uniform contraction of the centers does not increase (resp., decrease) the volume of the union (resp., intersection) of $N$ balls in Minkowski $d$-space, provided that $N\geq 2^d$ (resp., $N\geq 3^d$ and the unit ball of the Minkowski $d$-space is a generating set). Some improvements are presented in Euclidean spaces.

Hyperstability of a logarithmic-type functional equation on restricted domains

Let X be a real normed vector space and f:(0,∞)→X. In this paper, we prove the hyperstability of the logarithmic functional equation fxy−yf(x)=0on Γ of Lebesgue measure zero. More precisely, we prove that if f:(0,∞)→X satisfies ‖fxy−yf(x)‖≤ϕ(x,y)for all (x,y)∈Γ+⊂{(x,y):y>α(x)}[resp.Γ−⊂{(x,y):0<y<α(x)}] of Lebesgue measure zero, where α:(0,∞)→(0,∞) is an arbitrary given function and ϕ:(0,∞)×(0,∞)→[0,∞) satisfies the condition ϕ(x,y)∕y→0 as y→∞ [resp. y→0], then f satisfies the functional equation f(xy)=yf(x) for all x>0 andy>0.

Exact characterization for subdifferentials of a special optimal value function

For a closed set S and a bounded closed convex set U in a real normed vector space, we give exact subdifferential formulas of an optimal value function $$\mathrm {I}\!\Gamma _{S|U}$$ whose definition is based on the Minkowski function of U. $$\mathrm {I}\!\Gamma _{S|U}$$ covers distance function and indicator function as special cases. The main contribution is dropping two important assumptions of some main results in the literature.

Maxwell–Laman counts for bar-joint frameworks in normed spaces

The rigidity matrix is a fundamental tool for studying the infinitesimal rigidity properties of Euclidean bar-joint frameworks. In this paper we generalise this tool and introduce a rigidity matrix for bar-joint frameworks in arbitrary finite dimensional real normed vector spaces. Using this new matrix, we derive necessary Maxwell–Laman-type counting conditions for a well-positioned bar-joint framework in a real normed vector space to be infinitesimally rigid. Moreover, we derive symmetry-extended counting conditions for a bar-joint framework with a non-trivial symmetry group to be isostatic (i.e., minimally infinitesimally rigid). These conditions imply very simply stated restrictions on the number of those structural components that are fixed by the various symmetry operations of the framework. Finally, we offer some observations and conjectures regarding combinatorial characterisations of symmetric, isostatic bar-joint frameworks in (R2,‖⋅‖P), where the unit ball P is a quadrilateral.

On differentiability of mappings with finite dilation

Abstract We study mappings f : (a,b) → Y with finite dilation having Lebesgue integrable majorant, where Y is a real normed vector space. We construct Lipschitz mapping f : (a,b) → Y, dim Y = ∞ , which is nowhere differentiable but its graph has everywhere trivial contingent. We show that if the contingent of the graph of a mapping with finite dilation is a nontrivial space, then f is almost everywhere differentiable.

Uniform Domains and Uniform Domain Decomposition Property in Real Normed Vector Spaces

Let $E$ be a real normed vector space with $\dim(E)\geq 2$, $D$ a proper subdomain of $E$. In this paper we characterize uniform domains in $E$ in terms of the uniform domain decomposition property. In addition, we discuss the relation between quasiballs and domains with the quasiball decomposition property in $\mathsf{R}^n$.

Angles and a classification of normed spaces

We suggest a concept of generalized angles in arbitrary real normed vector spaces. We give for each real number a denition of an 'angle' by means of the shape of the unit ball. They all yield the well known Euclidean angle in the special case of real inner product spaces. With these dierent angles we achieve a classication of normed spaces, and we obtain a characterization of inner product spaces. Moreover we consider this construction also for a gener- alization of normed spaces, i.e. for spaces which may have a non-convex unit ball.

Decomposition properties of John domains in normed vector spaces

Let E be a real normed vector space with dim E ⩾ 2 . In this paper, we characterize John domains in E in terms of John domain decomposition property.

Unions of John domains and uniform domains in real normed vector spaces

Let E be real normed vector spaces with the dimension at least 2. In this paper we study the following questions: When is the union of two John domains in E a John domain and when is the union of two uniform domains in E a uniform domain? 1. Introduction and main results Throughout the paper, we always assume that E denotes a real normed vector space with dimE ‚ 2 and that D is a proper subdomain in E. The norm of a vector z in E is written as jzj, and for any two points z1;z2 in E, the distance between them is denoted by jz1 i z2j, and the closed line segment with endpoints z1 and z2 by (z1;z2). For x 2 E and r > 0, we let B(x;r) denote the open ball in E with center x and radius r. For real numbers r and s, we use the notation: r ^ s = minfr;sg. John domains in Euclidean spaces R n were introduced by John (1) in connection with his work on elasticity. The term is due to Martio and Sarvas (3). Roughly speaking, a domain is a John domain if it is possible to travel from one point of the domain to another without going too close to the boundary. The precise definition is as follows. Definition 1.1. D is called a c-John domain if for every pair of points x1;x2 2 D there is a rectifiable arc ∞ joining them with '(∞(x1;x)) ^ '(∞(x2;x)) • c d(x) for all x 2 ∞, where c is a positive constant, ∞(xj;x) denotes the closed subarc of ∞ with endpoints xj and x (j = 1;2), '(∞(xj;x)) the arclength of ∞(xj;x). ∞ is called a c-John arc joining x1 and x2. See (4) for several characterizations of John domains. In the study of John do-

Tangent cone and contingent cone to the intersection of two closed sets

For a nonempty closed set C in a real normed vector space X and an inequality solution set, we present several sufficient conditions for the tangent and contingent cones to their intersection to contain the intersections of the corresponding cones. We not only express the contingent cone to a solution set of inequalities and equalities by the directional (or Fréchet) derivatives of the active inequality constraint functions and the Fréchet derivatives of the equality constraint functions but also the tangent cone by the Clarke (or lower Dini, or upper Dini) derivatives of the active inequality constraint functions and the directional derivatives of the equality constraint functions. By using a simple property of the function d C − d C c , we characterize these cones by the hypertangent and hypercontingent vectors to the set C . Furthermore, these results allow us to present new constraint qualifications for the Karush–Kuhn–Tucker conditions.

The Quasiconvexity of Quasigeodesics in Real Normed Vector Spaces

In this paper, we assume that E is a real normed space of dimension at least two. The aim of this paper is to show that, for any convex domain in E, each quasigeodesic in this domain is quasiconvex in the norm metric. This result gives an affirmative answer to an open problem raised recently by Vaisala.

GENERALIZED F-IMPLICIT MULTIVALUED VARIATIONAL INEQUALITY PROBLEMS AND COMPLEMENTARITY PROBLEMS

In this paper, we study generalized F-implicit multivalued variational inequality problems on a real normed vector space setting. As an application, we study generalized F-implicit multivalued complementarity problems.

The convexity of the solution set of a pseudoconvex inequality

For a pseudoconvex function f on a nonempty convex set C in a real normed vector space X , we present several equivalent conditions for the convexity of the set C x ≔ { c ∈ C : f ( x ) ≤ f ( c ) } for x ∈ C . These conditions turn out to be very useful in characterizing the solution set of a pseudoconvex minimization problem of f over C x and the pseudolinearity of a Gâteux differentiable function f . We hence extend several existing results about characterizations of the solutions to a convex program and a pseudolinear program.

Regular hexagons in normed spaces and a theorem of Walter Benz

A new proof is given for the following recent result of W. Benz [1]. Suppose thatG: E → E, whereE is a real normed vector space of dimension at least 3, is such that, for somep o > 0, $$G(x + h) - G(x)\,and h\,\,are linearly dependent whenever\,\,\left\| h \right\| = p_0 .$$ Then there existx o ∈ E andq o ∈ ℝ such thatG(x) = q 0x + x0 for allx ∈ E. The new proof is based on existence theorems for regular hexagons in normed spaces which may be of some interest in themselves. Hexagons are also helpful in deducing similar results for certain two-dimensional spaces.

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.

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.

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.