Arbitrary Normed Space Research Articles

Existence and density theorems of Henig global proper efficient points

In this work, we provide some novel results that establish both the existence of Henig global proper efficient points and their density in the efficient set for vector optimization problems in arbitrary normed spaces. Our results do not require the assumption of convexity, and in certain cases, can be applied to unbounded sets. However, it is important to note that a weak compactness condition on the set (or on a section of it) and a separation property between the order cone and its conical neighbourhoods remains necessary. The weak compactness condition ensures that certain convergence properties hold. The separation property enables the interpolation of a family of Bishop-Phelps cones between the order cone and each of its conic neighbourhoods. This interpolation, combined with the proper handling of two distinct types of conic neighbourhoods, plays a crucial role in the proofs of our results, which include as a particular case other results that have already been established under more restrictive conditions.

Sufficient conditions for the preservation of polygonal-connectedness in an arbitrary normed space

Sufficient conditions for the preservation of polygonal-connectedness in an arbitrary normed space

Optimal Non-Gaussian Dvoretzky–Milman Embeddings

Abstract We construct the first non-gaussian ensemble that yields the optimal estimate in the Dvoretzky–Milman Theorem: the ensemble exhibits almost Euclidean sections in arbitrary normed spaces of the same dimension as the gaussian embedding—despite being very far from gaussian (in fact, it happens to be heavy-tailed).

Convergence of the Cimmino algorithm for common fixed point problems with a countable family of operators

In this paper we apply Cimmino algorithm for common fixed point problems with a countable family of quasi-nonexpansive operators in an arbitrary normed space and show its convergence. Our results are an extension of the recent results by T. Y. Kong , H. Pajoohesh and G. T. Herman obtained for operators which are projections on convex closed sets in a finite-dimensional Euclidean space.

On a theorem by Browder and its application to nonlinear boundary value problems

AbstractIn a paper from 1960, Felix Browder established a theorem concerning the continuation of the fixed points of a family of continuous functions depending continuously on a parameter , where is a convex and compact subset of . Here, the result is presented for a compact mapping where is a convex, closed, and bounded subset of an arbitrary normed space and is an arcwise connected topological space. Applications to nonlinear boundary value problems are given; specifically, we shall present new viewpoints of known results, introduce some novel results, and exhibit some open problems.

Symmetrized Birkhoff–James orthogonality in arbitrary normed spaces

Graph defined by Birkhoff–James orthogonality relation in normed spaces is studied. It is shown that (i) in a normed space of sufficiently large dimension there always exists a nonzero vector which is mutually Birkhoff–James orthogonal to each among a fixed number of given vectors, and (ii) in nonsmooth norms the cardinality of the set of pairwise Birkhoff–James orthogonal vectors may exceed the dimension of the vector space, but this cardinality is always bounded above by a function of the dimension. It is further shown that any given pair of elements in a normed space can be extended to a finite tuple such that each consecutive elements are mutually Birkhoff–James orthogonal; the exact minimal length of the tuple is also determined.

Tractability for Volterra problems of the second kind with convolution kernels

We study ε-approximation of the solution of the d-variateVolterra problem of the second kind, the Volterra operator having a convolution kernel. The Volterra operator is defined on an arbitrary normed function space Fd that is continuously embedded in the space of square integrable functions defined on the unit d-cube.Admissible information is given by continuous linear functionals on Fd. These functionals might be arbitrarily chosen; alternatively, we may allow only function values. (In the latter case, we can only consider spaces Fd for which function values are well-defined continuous linear functionals.) The error and cost are measured in the worst case or randomized setting in the L2-norm.Our first result is that lower and upper bounds on the information complexity of the Volterra problem and of d-variate L2-approximation differ by at most a constant factor. Using this result, we then show that necessary and sufficient conditions characterizing a given kind of tractability for multivariate approximation also characterize that same kind of tractability for the Volterra problem. We consider different kinds of algebraic tractabilities (in which we compare the information complexity to d and ε−1) and exponential tractability (in which we compare the information complexity to d and 1+lnε−1).However, our comparison of Volterra to approximation falls short in one respect. Multivariate approximation is linear; for many spaces (e.g., Hilbert spaces), the combinatory cost of multivariate approximation is roughly the same as its information complexity. But since the Volterra problem is nonlinear, it is unclear what the combinatory cost will be for the Volterra problem. This means that we do not know the extent to which the combinatory cost will exceed the information complexity. We partially address this issue by seeing whether the Picard iteration can give us an approximation without too great a penalty when we include the combinatory cost; this penalty is measured by the normalized combinatory cost, defined as the ratio of the combinatory cost to the cost of one admissible operation.In particular, suppose that we agree to compute an ε-approximation to the Volterra problem by (randomized) Monte Carlo. Suppose further that the convolution kernels are uniformly bounded in the L∞-norm. We then obtain an upper bound on the normalized combinatory cost of the Monte Carlo algorithm in the randomized setting. This upper bound is larger than the information complexity by roughly a factor of ε−2. This factor does not change positive results for algebraic tractability, but it does affect some of the positive results for exponential tractability.We also describe a deterministic algorithm that implements the Picard iteration. Assume that the approximation problem can be solved by linear algorithms that lie in a space of dimension proportional to the information complexity. We then find that the combinatory cost of the Picard iteration in the worst case setting is at most a power of the information complexity of d-variate approximation. This power is a constant if d−1lnε−1 is uniformly bounded. However, if d−1lnε−1 goes to infinity, then this power is of order (d−1lnε−1)∕ln(d−1lnε−1). Hence if d is large relative to lnε−1, this result is quite positive. On the other hand, for general d and ε, only some kinds of algebraic and exponential tractabilities hold when the normalized combinatory cost is included.

Vector valued multiplier spaces of $$f_\lambda $$-summability, completeness through $$c_0(X)$$-multiplier convergence and continuity and compactness of summing operators

Quite recently, the authors introduced the vector valued multiplier spaces associated to the series of bounded linear operators $$M^\infty _{f}\big (\sum _k T_k\big )$$ and $$M^\infty _{wf}\big (\sum _kT_k\big )$$ by means of almost and weak almost summability, respectively; [J. Math. Anal. Appl. 484: 123651]. As was recorded as an open problem in [J. Math. Anal. Appl. 484: 123651], in this study, we introduce vector valued multiplier spaces $$M^\infty _{f_\lambda }\big (\sum _k T_k\big )$$ and $$M^\infty _{wf_\lambda }\big (\sum _k T_k\big )$$ by means of generalized almost and weak almost summability, and give a characterization of completeness of these spaces, via $$c_0(X)$$ -multiplier convergent series. We also characterize the continuity and the (weak) compactness of the summing operator $${\mathcal {S}}$$ from the multiplier spaces $$M^\infty _{f_\lambda }\big (\sum _k T_k\big )$$ or $$M^\infty _{wf_\lambda }\big (\sum _k T_k\big )$$ to an arbitrary normed space Y through $$c_0(X)$$ -multiplier Cauchy and $$\ell _\infty (X)$$ -multiplier convergent series, respectively. Finally, we prove that if $$\sum _kT_k$$ is $$\ell _\infty (X)$$ -multiplier Cauchy, then the spaces $$M^\infty _{f_\lambda }\big (\sum _k T_k\big )$$ and $$M^\infty _{wf_\lambda }\big (\sum _k T_k\big )$$ are identical. These results are more general than the corresponding consequences given in [J. Math. Anal. Appl. 484: 123651] since almost convergence can be obtained from $$f_\lambda $$ -convergence under certain conditions.

Interpolation of Protoquantum Spaces

We consider the complex interpolation method in the context of protoquantum spaces. We apply it to construct an ℒp-protoquantization of an arbitrary normed space for 1 < p < ∞.

Relation of Pythagorean and Isosceles Orthogonality with Best approximations in Normed Linear Space

In an arbitrary normed space, though the norm not necessarily coming from the inner product space, the notion of orthogonality may be introduced in various ways as suggested by the mathematicians like R.C. James, B.D. Roberts, G. Birkhoff and S.O. Carlsson. We aim to explore the application of orthogonality in normed linear spaces in the best approximation. Hence it has already been proved that Birkhoff orthogonality implies best approximation and best approximation implies Birkhoff orthogonality. Additionally, it has been proved that in the case of ε -orthogonality, ε -best approximation implies ε -orthogonality and vice-versa. In this article we established relation between Pythagorean orthogonality and best approximation as well as isosceles orthogonality and ε -best approximation in normed space.

Operator valued series, almost summability of vector valued multipliers and (weak) compactness of summing operator

In this study, we introduce the vector valued multiplier spaces Mf∞(∑kTk) and Mwf∞(∑kTk) by means of almost summability and weak almost summability, and a series of bounded linear operators. Since these multiplier spaces are equipped with the sup norm and are subspaces of ℓ∞(X), we obtain the completeness of a normed space via the multiplier spaces which are complete for every c0(X)-multiplier Cauchy series. We also characterize the continuity and (weakly) compactness of the summing operator S from the multiplier spaces Mf∞(∑kTk) or Mwf∞(∑kTk) to an arbitrary normed space Y through c0(X)-multiplier Cauchy and ℓ∞(X)-multiplier convergent series, respectively. Finally, we show that if ∑kTk is ℓ∞(X)-multiplier Cauchy, then the multiplier spaces of almost convergence and weak almost convergence are identical. These results are more general than the corresponding consequences given by Swartz [20], and are analogues given by Altay and Kama [6].

Convexity of Suns in Tangent Directions

A direction $$d$$ is called a tangent direction to the unit sphere $$S$$ if the conditions that $$s \in S$$ and $${\text{lin}}(s + d)$$ is a supporting line to $$S$$ at the point s imply that $${\text{lin}}(s + d)$$ is a semitangent line to S, i.e., is the limit of secants at s. A set M is called convex in a direction $$d$$ if $$x,y \in M$$ and $$(y - x)\parallel d$$ imply that $$[x,y] \subset M$$ . In an arbitrary normed linear space, an arbitrary sun (in particular, a boundedly compact Chebyshev set) is proved to be convex in any tangent direction of the unit sphere.

Minimal convex majorants of functions and Demyanov–Rubinov exhaustive super(sub)differentials

ABSTRACTThe primary goal of the paper is to establish characteristic properties of (extended) real-valued functions defined on normed vector spaces that admit the representation as the lower envelope (the pointwise infimum) of their minimal (with respect of the pointwise ordering) convex majorants. The results presented in the paper generalize and extend the well-known Demyanov-Rubinov characterization of upper semicontinuous positively homogeneous functions as the lower envelope of exhaustive families of continuous sublinear functions to larger classes of (not necessarily positively homogeneous) functions defined on arbitrary normed spaces. As applications of the above results, we introduce, for nonsmooth functions, a new notion of the Demyanov-Rubinov exhaustive subdifferential at a given point, and show that it generalizes a number of known notions of subdifferentiability, in particular, the Fenchel-Moreau subdifferential of convex functions, the Dini-Hadamard (directional) subdifferential of directionally differentiable functions, and the Φ-subdifferential in the sense of the abstract convexity theory. Some applications of Demyanov-Rubinov exhaustive subdifferentials to extremal problems are considered.

Complexity and Approximation of Finding the Longest Vector Sum

The problem under study is, given a finite set of vectors in a normed vector space, find a subset which maximizes the norm of the vector sum. For each $${{\ell }_{p}}$$ norm, $$p \in [1,\infty )$$ , the problem is proved to have an inapproximability bound in the class of polynomial-time algorithms. For an arbitrary normed space of dimension $$O(logn)$$ , a randomized polynomial-time approximation scheme is proposed.

Multi-normed spaces based on non-discrete measures and their tensor products

Lambert discovered a new type of structures situated, in a sense, between normed spaces and abstract operator spaces. His definition was based on the notion of amplifying a normed space by means of the spaces . Later, several mathematicians studied more general structures (`-multi- normed spaces') introduced by means of the spaces , . We pass from to with an arbitrary measure. This becomes possible in the framework of the non- coordinate approach to the notion of amplification. In the case of a discrete counting measure, this approach is equivalent to the approach in the papers mentioned. Two categories arise. One consists of amplifications by means of an arbitrary normed space, and the other consists of -convex amplifications by means of . Each of them has its own tensor product of objects (the existence of each product is proved by a separate explicit construction). As a final result, we show that the `-convex' tensor product has an especially transparent form for the minimal -amplifications of -spaces, where is conjugate to . Namely, tensoring and , we obtain .

A new upper bound of geometric constant D(X)

A new constant mathit{WD}(X) is introduced into any real 2^{n}-dimensional symmetric normed space X. By virtue of this constant, an upper bound of the geometric constant D(X), which is used to measure the difference between Birkhoff orthogonality and isosceles orthogonality, is obtained and further extended to an arbitrary m-dimensional symmetric normed linear space (mgeq2). As an application, the result is used to prove a special case for the reverse Hölder inequality.

DIFFERENTIABILITY PROPERTIES OF ${\ell}$-STABLE VECTOR FUNCTIONS IN INFINITE-DIMENSIONAL NORMED SPACES

The aim of this paper is to continue the study of properties of an $\ell$-stable at a point vector function. We show that any $\ell$-stable at a point function from arbitrary normed linear space is strictly differentiable at the considered point.

APPROXIMATING FIXED POINTS OF NONEXPANSIVE TYPE MAPPINGS IN BANACH SPACES WITHOUT UNIFORM CONVEXITY

Approximate fixed point property problem for Mann iteration sequence of a nonexpansive mapping has been resolved on a Banach space independent of uniform (strict) convexity by Ishikawa [Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc. 59 (1976), 65-71]. In this paper, we solve this problem for a class of mappings wider than the class of asymptotically nonexpansive mappings on an arbitrary normed space. Our results generalize and extend several known results.

Alternative characterization of hyperbolic affine infinite iterated function systems

In this paper we present a characterization of hyperbolic affine infinite iterated function systems defined on an arbitrary normed space. Our result is a generalization of Theorem 1.1 from the paper “A characterization of hyperbolic affine iterated function systems”, Topology Proceedings, 36 (2010), 189–211, by R. Atkins, M. Barnsley, A. Vince and D. Wilson. Some examples are presented.

An asymptotically exact algorithm for the maximum traveling salesman problem in a finite-dimensional normed space

We consider the geometric maximum traveling salesman problem on assuming that the vertices of a graph lie in an arbitrary finite-dimensional normed space. For this problem we obtain an approximate algorithm with the relative error tending to zero as the number of vertices grows. The algorithm generalizes Serdyukov’s algorithm for the Euclidean Max TSP.