Asymmetric Normed Research Articles

Asymmetric Normed Baire Space

We prove that an asymmetric normed space is never a Baire space if the topology induced by the asymmetric norm is not equivalent to the topology of a norm. More precisely, we show that a biBanach asymmetric normed space is a Baire space if and only if it is isomorphic to its associated normed space.

The weak topology in finite dimensional asymmetric normed spaces

In this paper we show that, in contrast to what happens in a normed space, the weak topology of a finite dimensional asymmetric normed space is not necessarily the same as the topology of the asymmetric norm. We provide a class of finite dimensional asymmetric normed spaces where both topologies coincide. We also prove that the weak topology of an infinite dimensional asymmetric normed space is strictly coarser than the topology of the asymmetric norm.

On The Spaces of Linear Operators Acting Between Asymmetric Cone Normed Spaces

An asymmetric norm is a positive sublinear functional p on a real vector space X satisfying \(x=\theta _X\) whenever \(p(x)=p(-x)=0\). Since the space of all lower semi-continuous linear functionals of an asymmetric normed space is not a linear space, the theory is different in the asymmetric case. The main purpose of this study is to define bounded and continuous linear operators acting between asymmetric cone normed spaces. After examining the differences with symmetric case, we give some results related to Baire’s characterization of completeness in asymmetric cone normed spaces.

Local compactness in right bounded asymmetric normed spaces

We characterize the finite dimensional asymmetric normed spaces which are right bounded and the relation of this property with the natural compactness properties of the unit ball, such as compactness and strong compactness. In contrast with some results found in the existing literature, we show that not all right bounded asymmetric norms have compact closed balls. We also prove that there are finite dimensional asymmetric normed spaces that satisfy that the closed unit ball is compact, but not strongly compact, closing in this way an open question on the topology of finite dimensional asymmetric normed spaces. In the positive direction, we will prove that a finite dimensional asymmetric normed space is strongly locally compact if and only if it is right bounded.

Elementary approach to closed billiard trajectories in asymmetric normed spaces

We apply the technique of Károly Bezdek and Daniel Bezdek to study billiard trajectories in convex bodies, when the length is measured with a (possibly asymmetric) norm. We prove a lower bound for the length of the shortest closed billiard trajectory, related to the non-symmetric Mahler problem. With this technique we are able to give short and elementary proofs to some known results.

Compact convex sets in 2-dimensional asymmetric normed lattices

In this note, we study the geometric structure of compact convex sets in 2-dimensional asymmetric normed lattices. We prove that every q-compact convex set is strongly q-compact and we give a complete geometric description of the compact convex set with non empty interior in (ℝ2, q), where q is an asymmetric lattice norm.

A Characterization of Completeness via Absolutely Convergent Series and the Weierstrass Test in Asymmetric Normed Semilinear Spaces

Asymmetric normed semilinear spaces are studied. A description of biBanach, leftK-sequentially complete, and Smyth complete asymmetric normed semilinear spaces is provided and three appropriate notions of absolute convergence in the asymmetric normed framework are introduced. Some characterizations of completeness are also obtained via absolutely convergent series. Moreover, as an application, a Weierstrass test for the convergence of series is derived.

Ekeland Variational Principle in asymmetric locally convex spaces

In this paper we prove two versions of Ekeland Variational Principle in asymmetric locally convex spaces. The first one is based on a version of Ekeland Variational Principle in asymmetric normed spaces proved in S. Cobzaş, Topology Appl. 158 (8) (2011) 1073–1084. For the proof we need to study the completeness with respect to the asymmetric norm pA (the Minkowski functional) of the subspace XA of an asymmetric locally convex space X generated by a convex subset A of X (the analog of Banach disk). The second one is based on the existence of minimal elements (with respect to an appropriate order) in quasi-uniform spaces satisfying some completeness conditions, obtained as a consequence of Brezis–Browder maximality principle.

The Uniform Boundedness Theorem in Asymmetric Normed Spaces

We obtain a uniform boundedness type theorem in the frame of asymmetric normed spaces. The classical result for normed spaces follows as a particular case.

On aggregation of normed structures

In 1981, Borsik and Doboš studied in depth the problem of how to merge, by means of a function, a collection (not necessarily finite) of distance spaces in order to obtain a single one as a result [J. Borsik, J. Doboš, On a product of metric spaces, Math. Slovaca 31 (1981) 193–205]. Later on, Herburt and Moszyńska studied the same problem for the case of normed linear spaces, inspired by the fact that every norm induces in a natural way a distance on a linear space, and analyzed the relationship between the both aforenamed problems [I. Herburt, M. Moszyńska, On metric products, Colloq. Math. 62 (1991) 121–133]. More recently, Romaguera and Schellekens introduced a mathematical approach, based on the notions of asymmetric distance and asymmetric normed linear space, which is suitable for the complexity analysis of programs and algorithms in Computer Science [S. Romaguera, M. Schellekens, Quasi-metric properties of complexity spaces, Topology Appl. 98 (1999) 311–322]. In this paper, motivated by the importance of the information fusion techniques in Artificial Intelligence and by the utility of asymmetric distances and asymmetric norms in Computer Science, we study the Herburt and Moszyńska problem for asymmetric normed linear spaces. In particular we give a general description of how to combine a collection (not necessarily finite) of asymmetric normed linear spaces in order to obtain a single one as output and, in addition, we clear up the relationship between this problem and its analogous of combining asymmetric distance spaces which has been already explored by Mayor and Valero [G. Mayor, O. Valero, Aggregation of asymmetric distances in computer science, Inform. Sci. 180 (2010) 803–812]. Furthermore, it is shown that the asymmetric norms employed, in the spirit of Romaguera and Schellekens, in complexity analysis can be retrieved as a particular case of the developed theory. The last fact opens the possibility of applying a wide range of properties from the general aggregation theory in Artificial Intelligence to the complexity analysis of programs and algorithms in Computer Science.

The Goldstine Theorem for asymmetric normed linear spaces

It is well known that if ( X , q ) is an asymmetric normed linear space, then the function q s defined on X by q s ( x ) = max { q ( x ) , q ( − x ) } , is a norm on the linear space X. However, the lack of symmetry in the definition of the asymmetric norm q yields an algebraic asymmetry in the dual space of ( X , q ) . This fact establishes a significant difference with the standard results on duality that hold in the case of locally convex spaces. In this paper we study some aspects of a reflexivity theory in the setting of asymmetric normed linear spaces. In particular, we obtain a version of the Goldstine Theorem to these spaces which is applied to prove, among other results, a characterization of reflexive asymmetric normed linear spaces.

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.

Compactness in asymmetric normed spaces

A systematic study of precompact and compact subsets on asymmetric normed linear spaces is developed, centering our attention in the case of linear lattices with an asymmetric norm.

The supremum asymmetric norm on sequence algebras: A general framework to measure complexity distances

Recently, E.A. Emerson and C.S. Jutla (SIAM J. Comput., 1999), have successfully applied complexity of tree automata to obtain optimal deterministic exponential time algorithms for some important modal logics of programs. The running time of these algorithms corresponds, of course, to complexity functions which are potential functions and, thus, they do not belong, in general, to any dual p-complexity space.Motivated by these facts we here introduce and study a very general class of complexity spaces, which provides, in the dual context, a suitable framework to carry out a description of the complexity functions that generate exponential time algorithms. In particular, such spaces can be modelled as biBanach semialgebras which are isometrically isomorphic to the positive cone of the asymmetric normed linear space consisting of bounded sequences of real numbers endowed with the supremum asymmetric norm.

On the uniqueness of extension and unique best approximation in the dual of an asymmetric normed linear space

A well known result of R. R. Phelps (1960) asserts that in order that every linear continuous functional, defined on a subspace \(Y\) of a real normed space \(X\), have a unique norm preserving extension it is necessary and sufficient that its annihilator \(Y^\bot\) be a Chebyshevian subspace of \(X^\ast\). The aim of this note is to show that this result holds also in the case of spaces with asymmetric norm.