Linear operators in finite dimensional probabilistic normed spaces

In this paper, we consider strongly bounded linear operators on a finite dimensional probabilistic normed space and define the topological isomorphism between probabilistic normed spaces. Then we prove that every finite dimensional probabilistic normed space which is a topological vector space is complete.     

Quotient probabilistic normed spaces and completeness results

We introduce the concept of quotient in PN spaces and give some examples. We prove some theorems with regard to the completeness of a quotient.     

Generalized hessian forC 1,1 functions in infinite dimensional normed spaces

The subject of this paper is the systematic study of second order notions concerning differentiable functions with Lipschitz derivative. The results and notions are motivated by recent papers of Cominetti, Correa and Hiriart-Urruty. The first goal of this paper is the comparison of several known second order directional derivatives. The second goal is the introduction of a generalized Hessian which is a set of certain symmetric bilinear forms. The relation of this generalized Hessian to other existing second order derivatives is also described. The research was supported by a grant from the National Science Foundation NSF-66-2270, which is gratefully acknowledged. Research supported by the Hungarian National Foundation for Scientific Research (OTKA), Grant No. T-016846 and by the Humboldt Foundation.     

Statistical continuity in probabilistic normed spaces

In this study, we investigate the statistical continuity in a probabilistic normed space. In this context, the statistical continuity properties of the probabilistic norm, the vector addition and the scalar multiplication are examined.     

Some remarks on functions with values in probabilistic normed spaces

In this paper we consider an enlargement of the notion of the probabilistic normed space. For this new class of probabilistic normed spaces we give some topological properties. By using properties of the probabilistic norm we prove some differential and integral properties of functions with values into probabilistic normed spaces. As special cases, results for deterministic and random functions can be obtained.      

Statistically D-bounded sequences in probabilistic normed spaces

In this study, the concept of a statistically D-bounded sequence in a probabilistic normed (PN) space endowed with the strong topology is introduced and its basic properties are investigated. It is shown that a strongly statistically convergent sequence and a strong statistically Cauchy sequence are statistically D-bounded under certain conditions. A sequence which goes far away from the limit point infinitely many times and presents random deviations in a PN space may be handled with the tools of strong statistical convergence and statistical D-boundedness.     

Probabilistic normed Riesz spaces

In this paper, the concepts of probabilistic normed Riesz space and probabilistic Banach lattice are introduced, and their basic properties are studied. In this context, some continuity and convergence theorems are proved.     

Some topologies on a Šerstnev probabilistic normed space

In this paper we will introduce two other topologies, coarser than the so-called strong topology, on a class of Šerstnev probabilistic normed spaces, and obtain some important properties of these topologies. We will show that under the first topology, denoted by τ₀, our probabilistic normed space is decomposable into the topological direct sum of a normable subspace and the subspace of probably null elements. Under the second topology, which is in fact the inductive limit topology of a family of locally convex topologies, the dual space becomes a locally convex topological vector space.     

Functional inequalities in non-Archimedean normed spaces

In this paper, we introduce an additive functional inequality and a quadratic functional inequality in normed spaces, and prove the Hyers–Ulam stability of the functional inequalities in Banach spaces. Furthermore, we introduce an additive functional inequality and a quadratic functional inequality in non-Archimedean normed spaces, and prove the Hyers–Ulam stability of the functional inequalities in non-Archimedean Banach spaces.     

On ideal convergence of double sequences in probabilistic normed spaces

The notion of ideal convergence is a generalization of statistical convergence which has been intensively investigated in last few years.For an admissible ideal ∮N× N,the aim of the present paper is to introduce the concepts of ∮-convergence and ∮*-convergence for double sequences on probabilistic normed spaces(PN spaces for short).We give some relations related to these notions and find condition on the ideal ∮ for which both the notions coincide.We also define ∮-Cauchy and ∮*-Cauchy double sequences on PN spaces and show that ∮-convergent double sequences are ∮-Cauchy on these spaces.We establish example which shows that our method of convergence for double sequences on PN spaces is more general.     

Continuous operators on asymmetric normed spaces

If (X, p) and (Y, q) are two asymmetric normed spaces, the set LC(X, Y) of all continuous linear mappings from (X, p) to (Y, q) is not necessarily a linear space, it is a cone. If X and Y are two Banach lattices and p and q are, respectively, their associated asymmetric norms (p(x) = ‖⁺‖, q(y) = ‖y ⁺‖), we prove that the positive operators from X to Y are elements of the cone LC(X, Y). We also study the dual space of an asymmetric normed space and finally we give open mapping and closed graph type theorems in the framework of asymmetric normed spaces. The classical results for normed spaces follow as particular cases. The author acknowledges the support of the Ministerio de Educación y Ciencia of Spain and FEDER, under grant MTM2006-14925-C02-01 and Generalitat Valenciana under grant GV/2007/198.     

Weighted variational inequalities in normed spaces

In this article, we consider weighted variational inequalities over a product of sets and a system of weighted variational inequalities in normed spaces. We extend most results established in Ansari, Q.H., Khan, Z. and Siddiqi, A.H., (Weighted variational inequalities, Journal of Optimization Theory and Applications, 127(2005), pp. 263–283), from Euclidean spaces ordered by their respective non-negative orthants to normed spaces ordered by their respective non-trivial closed convex cones with non-empty interiors.     

Functions related to convexity and smoothness of normed spaces

We introduce a few functions related to convexity and smoothness of normed spaces. Those functions turn out to be moduli of convexity or smoothness or play an intermediate role. We calculate the exact formulas for introduced functions in some classical Banach spaces. An application to geometry of normed spaces is also indicated.     

Averaging distances in finite dimensional normed spaces and John's ellipsoid

A Banach space has the average distance property (ADP) if there exists a unique real number such that for each positive integer and all in the unit sphere of there is some in the unit sphere of such that

A theorem of Gross implies that every finite dimensional normed space has the average distance property. We show that, if has dimension , then . This is optimal and answers a question of Wolf (Arch. Math., 1994). The proof is based on properties of the John ellipsoid of maximal volume contained in the unit ball of .

     

Sufficient enlargements of minimal volume for finite-dimensional normed linear spaces

Let BY denote the unit ball of a normed linear space Y. A symmetric, bounded, closed, convex set A in a finite-dimensional normed linear space X is called a sufficient enlargement for X if, for an arbitrary isometric embedding of X into a Banach space Y, there exists a linear projection such that P(BY)⊂A. The main results of the paper: (1) Each minimal-volume sufficient enlargement is linearly equivalent to a zonotope spanned by multiples of columns of a totally unimodular matrix. (2) If a finite-dimensional normed linear space has a minimal-volume sufficient enlargement which is not a parallelepiped, then it contains a two-dimensional subspace whose unit ball is linearly equivalent to a regular hexagon.     

Projections of normed linear spaces with closed subspaces of finite codimension as kernels

Summary It follows from [1], [4] and [7] that any closed <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"11"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"12"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"13"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"14"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>n$-codimensional subspace ($n \ge 1$ integer) of a real Banach space $X$ is the kernel of a projection $X \to X$, of norm less than $f(n) + \varepsilon$~($\varepsilon > 0$ arbitrary), where \[ f (n) = \frac{2 + (n-1) \sqrt{n+2}}{n+1}. \] We have $f(n) < \sqrt{n}$ for $n > 1$, and \[ f(n) = \sqrt{n} - \frac{1}{\sqrt{n}} + O \left(\frac{1}{n}\right). \] (The same statement, with $\sqrt{n}$ rather than $f(n)$, has been proved in [2]. A~small improvement of the statement of [2], for $n = 2$, is given in [3], pp.~61--62, Remark.) In [1] for this theorem a deeper statement is used, on approximations of finite rank projections on the dual space $X^*$ by adjoints of finite rank projections on $X$. In this paper we show that the first cited result is an immediate consequence of the principle of local reflexivity, and of the result from [7].     

Common fixed point theorems in probabilistic metric spaces

We extend the notion of R-weak commutativity and its variants to probabilistic metric spaces and prove common fixed point theorems concerning them. Examples are included to reflect upon the distinctiveness of the types of mappings defined in the paper.