In this talk, we discuss an explicit method to define the linear structure of the Isbell-hull of an asymmetrically normed space

Conradie, Jurie

Künzi, Hans-Peter

Otafudu, Olivier Olela

English

University of Dayton

University of DaytoneCommonsSummer Conference on Topology and ItsApplications Department of Mathematics6-2017The Isbell-hull of an Asymmetrically NormedSpaceOlivier Olela OtafuduNorth-West University (South Africa), olivier.olelaotafudu@nwu.ac.zaJurie ConradieHans-Peter KünziFollow this and additional works at: http://ecommons.udayton.edu/topology_confPart of the Geometry and Topology Commons, and the Special Functions CommonsThis Topology + Asymmetric Structures is brought to you for free and open access by the Department of Mathematics at eCommons. It has beenaccepted for inclusion in Summer Conference on Topology and Its Applications by an authorized administrator of eCommons. For more information,please contact frice1@udayton.edu, mschlangen1@udayton.edu.eCommons CitationOtafudu, Olivier Olela; Conradie, Jurie; and Künzi, Hans-Peter, "The Isbell-hull of an Asymmetrically Normed Space" (2017).Summer Conference on Topology and Its Applications. 37.http://ecommons.udayton.edu/topology_conf/37OutlineIntroductionInjective asymmetrically normed spacesThe injective hull of an asymmetrically normed spaceThe Isbell-hull of an asymmetrically normedspaceOlivier Olela OtafuduSchool of Mathematical SciencesNorth-West University (Mafikeng Campus)University of Dayton, Dayton, Ohio, USAO. Olela Otafudu The Isbell-hull of an asymmetrically normed spaceOutlineIntroductionInjective asymmetrically normed spacesThe injective hull of an asymmetrically normed space1 Introduction2 Injective asymmetrically normed spaces3 The injective hull of an asymmetrically normed spaceO. Olela Otafudu The Isbell-hull of an asymmetrically normed spaceOutlineIntroductionInjective asymmetrically normed spacesThe injective hull of an asymmetrically normed spaceAsymmetrically normed spacesA function p : X → [0,∞) on a real vector space X will be called anasymmetric seminorm on X if for all x , y ∈ X and λ ∈ [0,∞),(a) p(λx) = λp(x);(b) p(x + y) ≤ p(x) + p(y).If in addition we also have(c) p(x) = p(−x) = 0 if and only if x = 0,p will be called an asymmetric norm, and the pair (X , p) an asymmetricallynormed space.If (a) is replaced by by:(a′) p(λx) = |λ|p(x) for all λ ∈ R, then p is called a semi-norm.If p is an asymmetric norm on X , the function pt : X → [0,∞) defined bypt(x) = p(−x), x ∈ Xis also an asymmetric norm, the asymmetric norm conjugate to p.The symmetrisation of the asymmetric norm p is the function ps : X → [0,∞)given byps(x) = max{p(x), p(−x)}, x ∈ Xand this is easily seen to be a norm on X .O. Olela Otafudu The Isbell-hull of an asymmetrically normed spaceOutlineIntroductionInjective asymmetrically normed spacesThe injective hull of an asymmetrically normed spaceAn asymmetric norm p induces a T0-quasi-metric dp on X defined bydp(x , y) = p(y − x) for all x , y ∈ X .For x ∈ X , r > 0 we define the ballsBpr (x) = {y ∈ X : dp(x , y) < r} = {y ∈ X : p(y − x) < r}andBpr [x ] = {y ∈ X : dp(x , y) ≤ r} = {y ∈ X : p(y − x) ≤ r}.The family {Bpr (x) : r > 0]} forms a fundamental system of neighbourhoods forx for a T0 topology τp on X , which we shall refer to as the topology induced byp.The Hausdorff topology τps induced by the norm ps is clearly finer than thetopologies τp and τpt .O. Olela Otafudu The Isbell-hull of an asymmetrically normed spaceOutlineIntroductionInjective asymmetrically normed spacesThe injective hull of an asymmetrically normed spaceExampleAs a simple but important example we mention the asymmetric norm p1 on R(regarded as a real vector space) defined for all x ∈ R byp1(x) = x+,where x+ = x ∨ 0 = max{x , 0} is the positive part of x . In this casept1(x) = max{−x , 0} = x−ps1(x) = max{x+, x−} = |x |.O. Olela Otafudu The Isbell-hull of an asymmetrically normed spaceOutlineIntroductionInjective asymmetrically normed spacesThe injective hull of an asymmetrically normed spaceDefinitionAn asymmetrically normed space (X , p) will be called (a) Isbell-convex if forevery family (xi )i∈I of elements of X and families of non-negative real numbers(ri )i∈I and (si )i∈I it follows that ifp(xj − xi ) ≤ ri + sjwhenever i , j ∈ I , then ⋂i∈IBpri [xi ] ∩ Bptsi [xi ] 6= ∅.(b) metrically convex if for every two elements x , y ∈ X and non-negativenumbers r and s such that p(y − x) ≤ r + s, there exists a z ∈ X such thatp(z − x) ≤ r and p(y − z) ≤ s.(c) Isbell-complete if for each family (xi )i∈I of elements in X and families ofnon-negative real numbers (ri )i∈I and (si )i∈I such that if Bpri [xi ] ∩ Bptsj [xj ] 6= ∅whenever i , j ∈ I , then ⋂i∈IBpri [xi ] ∩ Bptsi [xi ] 6= ∅.O. Olela Otafudu The Isbell-hull of an asymmetrically normed spaceOutlineIntroductionInjective asymmetrically normed spacesThe injective hull of an asymmetrically normed spaceExamples1 X = R2, ‖(x1, x2)‖ = max{|X1|, |X2|} has the binary intersection property.2 X = R2, ‖(x1, x2)‖ =√x21 + x22 does not have the binary intersectionproperty.3 Hence C, with its usual norm, does not have the binary intersectionproperty.O. Olela Otafudu The Isbell-hull of an asymmetrically normed spaceOutlineIntroductionInjective asymmetrically normed spacesThe injective hull of an asymmetrically normed spaceNote that an asymmetrically normed space (X , p) is Isbell-convex (metricallyconvex, Isbell-complete) if and only if the T0-quasi-metric space (X , dp) hasthe same property.LemmaEvery asymmetrically normed space (X , p) is metrically con- vex.LemmaAn asymmetrically normed space (X , p) is Isbell-convex if and only if it isIsbell-complete.O. Olela Otafudu The Isbell-hull of an asymmetrically normed spaceOutlineIntroductionInjective asymmetrically normed spacesThe injective hull of an asymmetrically normed spaceNote that an asymmetrically normed space (X , p) is Isbell-convex (metricallyconvex, Isbell-complete) if and only if the T0-quasi-metric space (X , dp) hasthe same property.LemmaEvery asymmetrically normed space (X , p) is metrically con- vex.LemmaAn asymmetrically normed space (X , p) is Isbell-convex if and only if it isIsbell-complete.O. Olela Otafudu The Isbell-hull of an asymmetrically normed spaceOutlineIntroductionInjective asymmetrically normed spacesThe injective hull of an asymmetrically normed spaceNote that an asymmetrically normed space (X , p) is Isbell-convex (metricallyconvex, Isbell-complete) if and only if the T0-quasi-metric space (X , dp) hasthe same property.LemmaEvery asymmetrically normed space (X , p) is metrically con- vex.LemmaAn asymmetrically normed space (X , p) is Isbell-convex if and only if it isIsbell-complete.O. Olela Otafudu The Isbell-hull of an asymmetrically normed spaceOutlineIntroductionInjective asymmetrically normed spacesThe injective hull of an asymmetrically normed spaceDefinitionAn asymmetrically normed space (Y , q) is called 1-injective if for everyasymmetrically normed space (X , p) and every linear subspace X0 of X , everycontinuous linear map T0 : (X0, p)→ (Y , q) has a continuous extensionT : X → Y such that ‖T‖p,q ≤ ‖T0‖p,q.LemmaIf the asymmetrically normed space (X , p) is Isbell-convex, then so is (X , pt),and the normed space (X , ps) is a hyperconvex Banach space, and therefore1-injective (as a Banach space).TheoremAn Isbell-convex (equivalently, Isbell-complete) asymmetrically normed space(Y , q) is 1-injective.O. Olela Otafudu The Isbell-hull of an asymmetrically normed spaceOutlineIntroductionInjective asymmetrically normed spacesThe injective hull of an asymmetrically normed spaceDefinitionAn asymmetrically normed space (Y , q) is called 1-injective if for everyasymmetrically normed space (X , p) and every linear subspace X0 of X , everycontinuous linear map T0 : (X0, p)→ (Y , q) has a continuous extensionT : X → Y such that ‖T‖p,q ≤ ‖T0‖p,q.LemmaIf the asymmetrically normed space (X , p) is Isbell-convex, then so is (X , pt),and the normed space (X , ps) is a hyperconvex Banach space, and therefore1-injective (as a Banach space).TheoremAn Isbell-convex (equivalently, Isbell-complete) asymmetrically normed space(Y , q) is 1-injective.O. Olela Otafudu The Isbell-hull of an asymmetrically normed spaceOutlineIntroductionInjective asymmetrically normed spacesThe injective hull of an asymmetrically normed spaceLet (X , p) be an asymmetrically normed space. Recall that a function pairf = (f1, f2), where fi : X → [0,∞) for i = 1, 2, is called ample ifp(y − x) ≤ f2(x) + f1(y),and that f is minimal whenever g = (g1, g2) is an ample pair such that ifg1(x) ≤ f1(x),g2(x) ≤ f2(x)for all x ∈ X , then g1 = f1, g2 = f2. The set of all minimal function pairs on Xwill be denoted by E(X , p). The following characterisation of the elements ofE(X , p) is useful; recall that for a ∈ R, we write a+ = a ∨ 0 = max{a, 0}.A function pair f = (f1, f2) belongs to E(X , p) if and only if for every x ∈ X ,f2(x) = sups∈X(p(s − x)− f1(s))+andf1(x) = sups∈X(p(x − s)− f2(s))+.O. Olela Otafudu The Isbell-hull of an asymmetrically normed spaceOutlineIntroductionInjective asymmetrically normed spacesThe injective hull of an asymmetrically normed spaceLet (X , p) be an asymmetrically normed space. Recall that a function pairf = (f1, f2), where fi : X → [0,∞) for i = 1, 2, is called ample ifp(y − x) ≤ f2(x) + f1(y),and that f is minimal whenever g = (g1, g2) is an ample pair such that ifg1(x) ≤ f1(x),g2(x) ≤ f2(x)for all x ∈ X , then g1 = f1, g2 = f2. The set of all minimal function pairs on Xwill be denoted by E(X , p). The following characterisation of the elements ofE(X , p) is useful; recall that for a ∈ R, we write a+ = a ∨ 0 = max{a, 0}.A function pair f = (f1, f2) belongs to E(X , p) if and only if for every x ∈ X ,f2(x) = sups∈X(p(s − x)− f1(s))+andf1(x) = sups∈X(p(x − s)− f2(s))+.O. Olela Otafudu The Isbell-hull of an asymmetrically normed spaceOutlineIntroductionInjective asymmetrically normed spacesThe injective hull of an asymmetrically normed spaceFor every z ∈ X , we define the minimal function pair fz = (fz,1, fz,2) byfz,1(x) = p(x − z), fz,2(x) = p(z − x).The mapping z 7→ fz is an injection of X into E(X , p).We now define scalar multiplication on E(X , p). For λ ∈ R and f ∈ E(X , p),we define the function pair f λ = (f λ1 , fλ2 ) byf λ1 (x) =λf1(λ−1x) if λ > 0,p(x) if λ = 0,|λ|f2(λ−1x) if λ < 0andf λ2 (x) =λf2(λ−1x) if λ > 0,p(−x) if λ = 0,|λ|f1(λ−1x) if λ < 0.O. Olela Otafudu The Isbell-hull of an asymmetrically normed spaceOutlineIntroductionInjective asymmetrically normed spacesThe injective hull of an asymmetrically normed spaceFor every z ∈ X , we define the minimal function pair fz = (fz,1, fz,2) byfz,1(x) = p(x − z), fz,2(x) = p(z − x).The mapping z 7→ fz is an injection of X into E(X , p).We now define scalar multiplication on E(X , p). For λ ∈ R and f ∈ E(X , p),we define the function pair f λ = (f λ1 , fλ2 ) byf λ1 (x) =λf1(λ−1x) if λ > 0,p(x) if λ = 0,|λ|f2(λ−1x) if λ < 0andf λ2 (x) =λf2(λ−1x) if λ > 0,p(−x) if λ = 0,|λ|f1(λ−1x) if λ < 0.O. Olela Otafudu The Isbell-hull of an asymmetrically normed spaceOutlineIntroductionInjective asymmetrically normed spacesThe injective hull of an asymmetrically normed spaceLemmaIf f = (f1, f2) ∈ E(X , p) and λ ∈ R, then f λ ∈ E(X , p)It now follows that we can define scalar multiplication in E(X , p) by puttingλf = f λ.We now turn to defining addition on E(X , p). Iff = (f1, f2), g = (g1, g2) ∈ E(X , p), x ∈ X we put f ⊕ g = ((f ⊕ g)1, (f ⊕ g)2),where(f ⊕ g)1(x) = sup{(f1(x − s)− g2(s))+ : s ∈ X},(f ⊕ g)2(x) = sup{(f2(x − s)− g1(s))+ : s ∈ X}.Lemma (Olela Otafudu, Topology Appl. 166 (2014))If f = (f1, f2) ∈ E(X , p), then for x ∈ X ,sup{(f1(x + s)− f1(s))+ : s ∈ X} = p(x),sup{(f2(x + s)− f2(s))+ : s ∈ X} = p(−x).O. Olela Otafudu The Isbell-hull of an asymmetrically normed spaceOutlineIntroductionInjective asymmetrically normed spacesThe injective hull of an asymmetrically normed spaceIf f , g ∈ E(X , p), then f ⊕ g is ample.Suppose x , y , z ∈ X . Then(fy ⊕ fz)1(x) = f(y+z),1(x)and(fy ⊕ fz)2(x) = f(y+z),2(x).LemmaIf f , g ∈ E(X , p) and x ∈ X , thensups∈X(f1(x − s)− g2(s))+ = sups∈X(g1(s)− f2(x − s))+andsups∈X(f2(x − s)− g1(s))+ = sups∈X(g2(s)− f1(x − s))+.O. Olela Otafudu The Isbell-hull of an asymmetrically normed spaceOutlineIntroductionInjective asymmetrically normed spacesThe injective hull of an asymmetrically normed spaceIf f , g ∈ E(X , p), then f ⊕ g is ample.Suppose x , y , z ∈ X . Then(fy ⊕ fz)1(x) = f(y+z),1(x)and(fy ⊕ fz)2(x) = f(y+z),2(x).LemmaIf f , g ∈ E(X , p) and x ∈ X , thensups∈X(f1(x − s)− g2(s))+ = sups∈X(g1(s)− f2(x − s))+andsups∈X(f2(x − s)− g1(s))+ = sups∈X(g2(s)− f1(x − s))+.O. Olela Otafudu The Isbell-hull of an asymmetrically normed spaceOutlineIntroductionInjective asymmetrically normed spacesThe injective hull of an asymmetrically normed spaceLet f , g , h ∈ E(X , p).Thenf ⊕ g = g ⊕ fandsuppose f ⊕ g , g ⊕ h ∈ E(X , p). Then(f ⊕ g)⊕ h = f ⊕ (g ⊕ h).In the light of the definition of scalar multiplication, the only candidate for theadditive identity is f 0 = (f 01 , f02 ), with f01 (x) = p(x), f02 (x) = p(−x)). We checkthis: Let f = (f1, f2) ∈ E(X , p) and x ∈ X . Then,(f 0 ⊕ f )1(x) = sup{(p(x − s)− f2(s))+ : s ∈ X} = f1(x),and(f 0 ⊕ f )2(x) = sup{(p(s − x)− f1(x))+ : s ∈ X} = f2(x).O. Olela Otafudu The Isbell-hull of an asymmetrically normed spaceOutlineIntroductionInjective asymmetrically normed spacesThe injective hull of an asymmetrically normed spaceLet f , g , h ∈ E(X , p).Thenf ⊕ g = g ⊕ fandsuppose f ⊕ g , g ⊕ h ∈ E(X , p). Then(f ⊕ g)⊕ h = f ⊕ (g ⊕ h).In the light of the definition of scalar multiplication, the only candidate for theadditive identity is f 0 = (f 01 , f02 ), with f01 (x) = p(x), f02 (x) = p(−x)). We checkthis: Let f = (f1, f2) ∈ E(X , p) and x ∈ X . Then,(f 0 ⊕ f )1(x) = sup{(p(x − s)− f2(s))+ : s ∈ X} = f1(x),and(f 0 ⊕ f )2(x) = sup{(p(s − x)− f1(x))+ : s ∈ X} = f2(x).O. Olela Otafudu The Isbell-hull of an asymmetrically normed spaceOutlineIntroductionInjective asymmetrically normed spacesThe injective hull of an asymmetrically normed spaceThe only candidate for the additive inverse of f = (f1, f2) is(−1)f = (f −11 , f −12 ). We check this:(f ⊕ f −1)1(x) = sup{(f1(x − s)− f −12 (s))+ : s ∈ X}= sup{(f1(x + s)− f1(s))+ : s ∈ X}= p(x) = f 01 (x).A similar calculation shows that(f ⊕ f −1)2(x) = p−1(x) = f 02 (x).We denote the additive inverse of f = (f1, f2) ∈ E(X , p) by −f ; thus−f = ((−f )1, (−f )2), where(−f )1(x) = f2(−x),(−f )2(x) = f1(−x)whenever x ∈ X . If f , g ∈ E(X , p), then f ⊕ g ∈ E(X , p).O. Olela Otafudu The Isbell-hull of an asymmetrically normed spaceOutlineIntroductionInjective asymmetrically normed spacesThe injective hull of an asymmetrically normed spaceTheoremIf scalar multiplication on E(X , p) is defined by λf = f λ and addition ⊕ by(f ⊕ g)1(x) = sup{(f1(x − s)− g2(s))+ : s ∈ X}and(f ⊕ g)2(x) = sup{(f2(x − s)− g1(s))+ : s ∈ X}whenever f , g ∈ E(X , p) and λ ∈ R, then E(X , p) is a vector space and themap x 7→ fx is a linear isomorphism of X into E(X , p).O. Olela Otafudu The Isbell-hull of an asymmetrically normed spaceOutlineIntroductionInjective asymmetrically normed spacesThe injective hull of an asymmetrically normed spaceTo define an asymmetric norm on E(X , p) we take our cue from theT0-quasi-metric D defined on the injective hull of a T0-quasi-metric space(X , d) in [Kemajou, Ku¨nzi, Otafudu, Topology Apppl. 159 (2012)] byD(f , g) = max{sups∈X(f1(s)− g1(s))+, sups∈X(g2(s)− f2(s))+}for f , g ∈ E(X , p), it is shown thatD(f , g) = sups∈X(f1(s)− g1(s))+ = sups∈X(g2(s)− f2(s))+.Recall that the additive identity f 0 = (f 01 , f02 ) is defined byf 01 (s) = p(s), f02 (s) = p(−s).For f ∈ E(X , p) we now putp˜(f ) = D(f 0, f ) = sups∈X(f2(s)− p(−s))+ = sups∈X(p(x)− f1(s))+ = f2(0)andp˜(−f ) = D(f , f 0) = f1(0).O. Olela Otafudu The Isbell-hull of an asymmetrically normed spaceOutlineIntroductionInjective asymmetrically normed spacesThe injective hull of an asymmetrically normed spaceThe function p˜ : E(X , p)→ [0,∞) defined above is an asymmetric norm onE(X , p) and the map x 7→ fx is an isometry.TheoremAn 1-injective asymmetrically normed space (X , p) is Isbell-convex.LemmaE(X , p) is Isbell-convex.O. Olela Otafudu The Isbell-hull of an asymmetrically normed spaceOutlineIntroductionInjective asymmetrically normed spacesThe injective hull of an asymmetrically normed spaceThe function p˜ : E(X , p)→ [0,∞) defined above is an asymmetric norm onE(X , p) and the map x 7→ fx is an isometry.TheoremAn 1-injective asymmetrically normed space (X , p) is Isbell-convex.LemmaE(X , p) is Isbell-convex.O. Olela Otafudu The Isbell-hull of an asymmetrically normed spaceOutlineIntroductionInjective asymmetrically normed spacesThe injective hull of an asymmetrically normed spaceThe function p˜ : E(X , p)→ [0,∞) defined above is an asymmetric norm onE(X , p) and the map x 7→ fx is an isometry.TheoremAn 1-injective asymmetrically normed space (X , p) is Isbell-convex.LemmaE(X , p) is Isbell-convex.O. Olela Otafudu The Isbell-hull of an asymmetrically normed spaceOutlineIntroductionInjective asymmetrically normed spacesThe injective hull of an asymmetrically normed spaceE. Kemajou, H.-P. Ku¨nzi and O. Olela Otafudu, The Isbell-hull of adi-space, Topology Appl. 159 (2012) 2463–2475.J.R. Isbell, Six theorems about injective metric spaces, Comment. Math.Helv. 39 (1964), 65–76.O. Olela Otafudu, Extremal function pairs in asymmetric normed linearspaces, Topology Appl. 166 (2014) 98-107.O. Olela Otafudu The Isbell-hull of an asymmetrically normed space

The Isbell-hull of an Asymmetrically Normed Space

https://ecommons.udayton.edu/cgi/viewcontent.cgi?article=1045&amp;context=topology_conf

The Isbell-hull of an Asymmetrically Normed Space

Abstract

Similar works

Full text

Available Versions

University of Dayton