Normed Linear Spaces Research Articles

The mapping in normed linear spaces with applications to inequalities in analysis

The mapping in normed linear spaces with applications to inequalities in analysis

On the optimal renewal of bilinear functionals in linear Normed spaces

We study the problem of optimal renewal of bilinear functionals on the basis of optimal linear information in the general statement. We also represent some new results for special spaces of functions.

A Bernstein–Markov Theorem for Normed Spaces

LetXandYbe real normed linear spaces and let φ:X→R be a non-negative function satisfying φ(x+y)≤φ(x)+‖y‖ for allx,y∈X. We show that there exist optimal constantscm,ksuch that ifP:X→Yis any polynomial satisfying ‖P(x)‖≤φ(x)mfor allx∈X, then ‖D̂kP(x)‖≤cm,kφ(x)m−kwheneverx∈Xand 0≤k≤m. We obtain estimates for these constants and present applications to polynomials and multilinear mappings in normed spaces.

Pythagorean euclidean four point properties

Characterizations of real inner product spaces among a class of metric spaces have been obtained based on homogeneity of metric pythagorean orthogonality, a metrization of the concept of pythagorean orthogonality as defined in normed linear spaces. In the present paper a considerable weakening of this hypothesis is shown to characterize real inner product spaces among complete, convex, externally convex metric spaces, generalizing a result of Kapoor and Prasad [9], and providing a connection with the many characterizations of such spaces using euclidean four point properties.

Area orthogonality in normed linear spaces

A new concept of orthogonality in real normed linear spaces is introduced. Typical properties of orthogonality (homogeneity, symmetry, additivity, ...) and relations between this orthogonality and other known orthogonalities (Birkhoff, Boussouis, Unitary-Boussouis and Diminnie) are studied. In particular, some characterizations of inner product spaces are obtained.

PARTIALLY ORDERED NORMED lINEAR SPACES WITH WEAK FATOU PROPERTY

Let $E$ be a Riesz space with lattice ordered norm $\|\cdot\|$. Amemiya proved that $E$ is complete under this norm if $E$ has weak Fatou property for monotone sequence (: $E$ is monotone complete) with respect to the norm $\|\cdot\|$. This is a generalization of the Riesz-Fisher's and the Nakano's theorem. In the cases of non normed Riesz space or non lattice ordered norm, this theorem is not true in general. We shall investigate in this paper a necessary and sufficient condition for Amemiya's theorem to be valid in a partially ordered normed linear space.

Sub-exponential Solutions of Multidimensional DifferenceEquations

We shall prove that practically any recursive solution of a linear multidimensional difference equation increases sub-exponentially. If all the steps of the recursive procedure are ’’of the same direction‘‘, useful growth estimates can be given. The results are also generalized to some cases of non-linear difference equations and sequences in the linear normed space.

On the Cauchy difference on normed spaces

LetX be a real linear normed space, (G, +) be a topological group, andK be a discrete normal subgroup ofG. We prove that if a continuous at a point or measurable (in the sense specified later) functionf:X →G fulfils the condition:f(x +y) -f(x) -f(y) ∈K whenever ‖x‖ = ‖y‖, then, under some additional assumptions onG,K, andX, there esists a continuous additive functionA :X →G such thatf(x) -A(x) ∈K.

Notes on weighted norms and network approximation of functionals

Among other results in the literature concerning arbitrarily good approximation that concern more general types of target functionals, different network structures, other nonlinearities, and various measures of approximation errors is the proposition that any continuous real nonlinear functional on a compact subset of a real normed linear space can be approximated arbitrarily well using a single-hidden-layer neural network with a linear functional input layer and exponential (or polynomial or sigmoidal or radial basis function) nonlinearities.

Best Interpolatory Approximation in Normed Linear Spaces

A theory of best approximation with interpolatory contraints from a finite-dimensional subspaceMof a normed linear spaceXis developed. In particular, to eachx∈X, best approximations are sought from a subsetM(x) ofMwhichdependson the elementxbeing approximated. It is shown that this “parametric approximation” problem can be essentially reduced to the “usual” one involving a certainfixedsubspaceM0ofM. More detailed results can be obtained when (1)Xis a Hilbert space, or (2)Mis an “interpolating subspace” ofX(in the sense of [1]).

Weak additivity of metric pythagorean orthogonality

It is well known that the property of additivity of pythagorean orthogonality characterizes real inner product spaces among normed linear spaces. In the present paper, a natural concept of additivity is introduced in metric spaces, and it is shown that a weakened version of this additivity of metric pythagorean orthogonality characterizes real inner product spaces among complete, convex, externally convex metric spaces, providing a generalization of the earlier characterization.

Extension Theorems of Continuous Random Linear Operators on Random Domains

The central purpose of this paper is to prove the following theorem: let (Ω, σ, u) be a complete probability space, ( B, ∥·∥) a normed linear space over the scalar field K, E: Ω → 2 B a separable random domain with linear subspace values, and ƒ: Gr E → K a continuous random linear operator, where Gr E = {(ω, x) ∈ Ω × B| x ∈ E(ω)} denotes the graph of E. Then there exists a continuous random linear operator ƒ̃: Ω × B → K such that ƒ̃(ω, x) = ƒ(ω, x) ∀ ω ∈ Ω, x ∈ E(ω), and sup{|ƒ̃(ω, x)| | x ∈ B, ∥ x∥ ≤ 1} = sup{|ƒ(ω, x)| | x ∈ E(ω), ∥ x∥ ≤ 1}, for each ω in Ω. For the case where E is not separable, a result similar to the above-stated theorem is also given, which generalizes and improves many previous results on random generalizations of the Hahn-Banach Theorem.

An Optimal Krasnosel'skii-Type Theorem for the Dimension of the Kernel of a Starshaped Set

It is shown for a closed connected nonconvex and locally compact subset S of a real normed linear space that ker S = ∩ {conv Sz : z ∈ D ∩ reg S}, where reg S denotes the set of regular points of S, D is a relatively open subset of S containing the set lnc S of local nonconvexity points of S, and Sz = {s ∈ S : z is visible from s via S}. An analogous intersection formula, with the set sph S of spherical points of S in place of reg S, is shown to hold for a closed connected nonconvex set S in a real Banach space which is uniformly convex and uniformly smooth. A routine procedure then leads to a Krasnosel'skii-type characterization of the dimension of the kernel of a closed connected nonconvex subset S of Rd with lnc S bounded. This improves a recent result by removing the hypothesis of compactness of S and settles an open problem.

Sums of distances in normed spaces

Abstract : A geometric proof for the following theorem due to Martelli and Busenberg is given. Integral geometry is used to discuss special cases and related results. Minkowski spaces are simply finite dimensional normed linear spaces. Smoothness assumptions on the boundary of the unit disk E for a Minkowski plane will enable us to use Crofton's simplest formula from integral geometry to give a proof for three points. If the unit ball for a 3-dimensional Minkowski space is a zonoid, then we used integral geometry for the case of four points forming a simplex. A zonoid is a limit of sums of segments. Bolker discusses equivalent conditions for a convex subset of Rn to be a zonoid.

ORDERS OF MODULI OF CONTINUITY OF OPERATORS OF ALMOST BEST APPROXIMATION

Let be a normed linear space, a finite-dimensional subspace, . A multiplicative -selection , where , is a single-valued mapping such that It is proved in the paper that when , , for any and there exists an -selection such that where the estimate is order-sharp in the space . It is also established that the Lipschitz constant for the -selection is of proximate order in the spaces and . Bibliography: 21 titles.

Classifiers on relatively compact sets

The problem of classifying signals is of interest in several application areas. Typically we are given a finite number m of pairwise disjoint sets C/sub 1/, ..., C/sub m/ of signals, and we would like to synthesize a system that maps the elements of each C/sub j/ into a real number a/sub j/, such that the numbers a/sub 1/, ..., a/sub m/ are distinct. In a recent paper it is shown that this classification can be performed by certain simple structures involving linear functionals and memoryless nonlinear elements, assuming that the C/sub j/ are compact subsets of a real normed linear space. Here we give a similar solution to the problem under the considerably weaker assumption that the C/sub j/ are relatively compact and are of positive distance from each other. An example is given in which the C/sub j/ are subsets of L/sub p/(a,b), 1/spl les/p >

Continuous Selections for Set-Valued Mappings

A theorem is proved which states that any almost lower semicontinuous set-valued mapping with convex and closed images from a paracompact topological to a one-dimensional normed linear space admits a continuous selection.

General structures for classification

The problem of classifying signals is of interest in several application areas. Typically, we are given a finite number m of pairwise disjoint sets C/sub 1/,...,C/sub m/ of signals, and we would like to synthesize a system that maps the elements of each C/sub j/ into a real number a/sub j/, such that the numbers a/sub 1/,...,a/sub m/ are distinct. The main purpose of this paper is to show that this classification can be performed by certain simple structures. Involving linear functionals and memoryless nonlinear elements, assuming only that the C/sub j/ are compact subsets of a real normed linear space. The results on which this conclusion is based have applications other than to classification problems. For example, one result provides a relatively simple completion of a proof of a well-known proposition concerning approximations in R/sup n/ using sigmoidal functions. >

Some propositions related to a dilation theorem of W. Benz

This paper begins with another proof of a theorem of W. Benz [2] concerning dilations in normed linear spaces. Our proof motivates several questions which are addressed thereafter. For instance it is shown that, ifI is an open interval in ℝ,γ: I → ℝ n ,γ is continuously differentiable and there exista1,...,a n ∈I such that {γ′(a1,...,γ′(a n )} is linearly independent, then {γ(t): t ∈ I} contains a Hamel basis for ℝ n over ℚ.

Lipschitz Regularization and the Convergence of Convex Functions

Let be a normed linear space,and f be a proper lower semicontinuous convex function defined on X. The Lipschitz reg-ularization of f with parameter μ is the infimal convolution of f with , defined by . In this article we express, in an arbitrary normed space, the twomost important epigraphical convergence notions for convex functions in terms of the convergence of associated Lipschitz regularizations with respect to classical function theoretic modes of convergence. In the case of slice convergence of epigraphs, the associated mode of convergence for the regularizations is pointwise convergence, whereas for Attouch-Wets convergence of epigraphs, the associated mode of convergence for the regularizations is uniform convergence on bounded subsets of XL.