Global Journal of Mathematics and Mathematical Sciences (GJMMS). ISSN 0972-9836 Volume 6, Number 1 (2016), pp. 1–9 © Research India Publications http://www.ripublication.com/gjmms.htm Decomposition of metrics and norms Hernández J. M., Castañeda C. H., Álvarez L. C., Tochihuitl V. and Vázquez R. Instituto de Física y Matemáticas, Universidad Tecnológica de la Mixteca. E-mail: [email protected], [email protected], [email protected], [email protected], [email protected] Abstract In this paper we describe as give a decomposition of a metric (norm) in two con- jugated quasi-metrics (asymmetric norm, respectively). After we introduce the notions of quasi-k-metrics and asymmetric k-norms and, under certain assump- tions also we give different ways to decompose a metric (norm) in two conjugated quasi-k-metrics (asymmetric k-norms, respectively). Finally we illustrate these with examples. AMS Subject Classification: 46B20, 54E25, 54E55. Keywords: Asymmetric norm, quasi-metric, quasi-k-metric, asymmetric k-norm. 1. Introduction It is well known that, if one attempts to omit the requirement of symmetry in a semimetric or in a seminorm, then the structure of the space is drastically changed. Such unsymmet- rical spaces have been studied by Wilson [7] who used the term quasi-semimetric when the feature of symmetry is omitted in a semimetric, and Duffin and Karlovitz (1968) [4] who proposed the term asymmetric seminorm when the property of symmetry is not satisfied in a seminorm. Subsequently, many classical results of the analysis have been extended to such non-symmetric spaces, see for example [1], [5], [6], [8] and in [2] a compendium of works performed. Formally, a quasi-semimetric on a set X is a mapping ρ : X ×X −→ [ 0, ∞) satisfying the following conditions: (i) ρ(x,x) = 0, (ii) ρ(x,z) ≤ ρ(x,y) + ρ(y,z), for all x,y,z ∈ X. If, further, (iii) ρ(x,y) = ρ(y,x) = 0 implies x = y, then ρ is called a quasi-metric; while for a linear space X a func- tion p : X → R is an asymmetric norm if for all x,y ∈ X and yr ∈[0, ∞), (i) 2 Hernández J. M., Castañeda C. H., Álvarez L. C., Tochihuitl V. and Vázquez R. p(x) = p(−x) = 0 ⇒ x = 0, (ii) p(rx) = rp(x) and (iii) p(x + y) ≤ p(x) + p(y). If only (ii) and (iii) are satisfied, then p is called an asymmetric seminorm. A quasi- semimetric ρ determines another quasi-semimetric ρ defined by ρ(x,y) = ρ(y,x) and also a semimetric ρs given by ρs = max{ρ,ρ}; similarly, an asymmetric seminorm determines its conjugated asymmetric seminorm p(x) = p(−x) and its associated semi- norm ps = max{p, p}. Here we treat the problem: given a metric d (a norm ·), find a quasi-metric ρ (asymmetric norm p) such that d = ρs (· = ps), then we introduce the definitions of quasi-k-metric and asymmetric-k-norm with which, we try the similar problem: given a metric d (a norm ·), we find a quasi-k-metric ρ (asymmetric-k-norm p) with d = ρs (· = ps).It is good to mention that in [3] the author defines that an asymmetric normed linear space (E, q) is normable if there is a norm ||·||on the linear space E such that the topologies τ dq and τ d||·|| coincide on E. This is different from what we do in this work.1 2. Definitions In the following definitions, k is a real number greater than or equal to 1. Definition 2.1. Let X be a nonempty set, a k-metric on X is a function d : X × X → [0, ∞) which satisfies for all x,y,z ∈ X, the conditions: (i) d(x,y) = 0 if and only if x = y, (ii) d(x,z) ≤ k (d(x,y) + d(y,z)) , (iii) d(x,y) = d(y,x). The pair (X, d) is called a k-metric space. If only (ii) and (iii) are satisfied, then the pair is called a k-semimetric space. Definition 2.2. Let X be a real linear space. A function · k : X →[0, ∞) is a k-norm on X if for each x,y ∈ X and r ∈ R : (i) x k = 0 if and only if x = 0, | | (ii) rx k = r x k , + ≤ + (iii) x y k k x k y k . (X, · k) is called a k-normed space. Also if only (ii) and (iii) are satisfied, then the pair is called a k-seminormed space. Remark 2.3. A metric (semimetric) space is a 1-metric space (1-semimetric space) and a normed (seminormed) space is a 1-normed space (1-seminormed space). 1This work is partially supported by Programa de Mejoramiento del Profesorado (UTMIX-EXB-025). Decomposition of metrics and norms 3 Definition 2.4. A quasi-k-semimetric on an arbitrary nonempty set X is a function ρ : X × X →[0, ∞) that satisfies: (i) ρ(x,x) = 0, (ii) ρ(x,z) ≤ k (ρ(x,y) + ρ(y,z)) , for all x,y,z ∈ X. If, further, (iii) ρ(x,y) = ρ(y,x) = 0 implies x = y, for all x,y ∈ X, then ρ is called a quasi-k-metric. A quasi-metric (quasi-semimetric) is a quasi-1-metric (quasi-1-semimetric). x if x ≥ 0 R = Example 2.5. is a quasi-2-metric space, with ρ(x,y) x . − if x<0 2 Like a quasi-semimetric, given a quasi-k-semimetric ρ in a set X, the conjugate ρ of ρ is defined by ρ(x,y) = ρ(y,x), x,y ∈ X. It is easy to verify that ρ is also a quasi-k-semimetric, meanwhile the function ρs : X × X → R given by ρs(x, y) = max {ρ(x,y),ρ(x,y)} , (2.1) provestobeak-semimetric. Even further, ρ is a quasi-k-metric if and only if ρs is a k-metric on X. The following inequality is trivially valid, ρ(y,x) ≤ ρs(x, y) and ρ(x,y) ≤ ρs(x, y). (2.2) Definition 2.6. Let X be a real linear space. A function p : X →[0, ∞) is an asymmetric k-norm on X, if for all x,y ∈ X and r ≥ 0, (i) p(x) = p(−x) = 0 implies x = 0; (ii) p(rx) = rp(x); (iii) p(x + y) ≤ k (p(x) + p(y)) . In this case (X, p) is called an asymmetrically k-normed space. If p only satisfied the conditions (ii) and (iii), then it is called an asymmetric k-seminorm, and the pair (X, p) is called an asymmetrically k-seminormed space. Also in some instances the value +∞ is allowed to p, in which case p is an extended asymmetric k-seminorm (k-seminorm) The conjugate p of the asymmetric k-seminorm p is defined by p(x) = p(−x), x ∈ X. 4 Hernández J. M., Castañeda C. H., Álvarez L. C., Tochihuitl V. and Vázquez R. In analogy to (2.1), the function ps : X →[0, ∞) given by ps(x) = max{p(x), p(x)},x∈ X, is a k-seminorm. An asymmetric k-seminorm p is an asymmetric k-norm if and only if ps is a k-norm on X. An asymmetric k-seminorm p induces a quasi-k-semimetric ρp on X, given by = − ∈ ρp(x, y) p(y x), x,y X. Here the inequalities in (2.2) become p(x) ≤ ps(x) and p(x) ≤ ps(x) for all x ∈ X. Definition 2.7. If d is a metric on X and for some quasi-k-metric ρ = d is satisfied that ρs = d, then the pair (ρ,ρ) is called a decomposition of the metric d. Similarly, if · is a norm on X and p = · is an asymmetric k-norm such that ps = · , then the pair (p, p) is called a decomposition of the norm · . 3. Results Now, we give some sufficient conditions to ensure that a metric space comes from a quasi-metric space. Theorem 3.1. Let (X, d) be a metric space. If f : X → R is a function satisfying d(x,z) ≥ d(y,z) and d(x,z) ≥ d(x,y) whenever f(x) ≤ f(y) ≤ f(z), then there exists a decomposition of d. d(x,y) if f(x)≤ f(y) Proof. Let ρ(x,y) = ρ (x, y) = . Let us prove that ρ is a f 0iff(x)>f(y) quasi-metric, we will start showing the triangular inequality by cases: given x,y,z ∈ X Case 1 f (x) > f (y). ρ(x,y) = 0 ≤ ρ(x,z) + ρ(z,y). Case 2 f(x)≤ f(y).There are 3 subcases. Subcase 1: f(x)≤ f (y) < f (z). ρ(x,z) + ρ(z,y) = d(x,z) ≥ d(x,y) = ρ(x,y). Subcase 2: f(x)≤ f(z)≤ f(y). ρ(x,y) = d(x,y) ≤ d(x,z) + d(z,y) = ρ(x,z) + ρ(z,y). Subcase 3: f(z)<f(x)≤ f(y). ρ(x,z) + ρ(z,y) = d(z,y) ≥ d(x,y) = ρ(x,y). Decomposition of metrics and norms 5 In addition, it is clear that ρ(x,y) ≥ 0, and if ρ(x,y) = ρ(y,x) = 0, then d(y,x) = 0 and so x = y. _ d(y,x) if f(y)≤ f(x) Furthermore, ρ(x,y) = ρ(y,x) = 0 if f(y)>f(x) thus, ρs(x, y) = max {ρ(x,y),ρ(y,x)} = d(x,y).