Quantified functional analysis: recapturing the dual unit ball

Sioen, Mark; Verwulgen, Stijn

doi:10.1007/bf03323389

Cited by 5 publications

(15 citation statements)

References 11 publications

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“…First, the topological coreflection of the Mackey structure is equal to the Mackey topology, of the coreflection of the original structure. Secondly, it follows from [12] that the Mackey structure of a topological locally convex approach space is topological and thus equal to the Mackey topology. Last, as a result of the HahnYBanach theorem, a seminorm equals its own Mackey structure, which numerifies the fact that the Mackey topology of a seminormable topology is invariant.…”

Section: Limits In Dualitymentioning

confidence: 98%

“…We refer to [12] for a comprehensive account on the information below. The set of linear contractions on a given locally convex approach space ðX ; MÞ, i.e., those linear functionals ' : X !…”

Section: Preliminariesmentioning

confidence: 99%

“…They, moreover, work well in functional analysis [7,8,11], and in [12] and [10] it is pointed out that the dual of a locally convex approach space is in a natural way a seminormed space. So locally convex approach spaces might form a suitable category to settle the above question.…”

Section: Introductionmentioning

confidence: 99%

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An Isometric Representation of the Dual of $ {\user1{\mathcal{C}}}{\left( {X,\mathbb{R}} \right)} $

Verwulgen

2006

Appl Categor Struct

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Many structures in functional analysis are introduced as the limit of an inverse (aka projective) system of seminormed spaces [2,4,9]. In these situations, the dual is moreover equipped with a seminorm. Although the topology of the inverse limit is seldom metrizable, there is always a natural overlying locally convex approach structure. We provide a method for computing the adjoint of this space, by showing that the dual of a limit of locally convex approach spaces becomes a co-limit in the category of seminormed spaces. As an application we obtain an isometric representation of the dual space of real valued continuous functions on a locally compact Hausdorff space X, equipped with the compact open structure. (2000): 46M40, 46A03, 46E27, 28A33. Mathematics Subject Classifications

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Section: Limits In Dualitymentioning

confidence: 98%

Section: Preliminariesmentioning

confidence: 99%

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An Isometric Representation of the Dual of $ {\user1{\mathcal{C}}}{\left( {X,\mathbb{R}} \right)} $

Verwulgen

2006

Appl Categor Struct

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“…The structural flexibility of approach theory entails the existence of canonical approach spaces in branches of mathematical analysis such as functional analysis ( [LS00], [SV03], [LV04], [SV04], [SV06], [SV07]), hyperspace theory ( [LS96], [LS98],[LS00']), domain theory ( [CDL11], [CDL14], [CDS14]), and probability theory and statistics ( [BLV11], [BLV13], [BLV16]). A careful study of these approach spaces has resulted in new insights and applications in these branches.…”

Section: Introductionmentioning

confidence: 99%

An application of approach theory to the relative Hausdorff measure of non-compactness for the Wasserstein metric

Berckmoes¹,

Hellemans²,

Sioen³

et al. 2017

Journal of Mathematical Analysis and Applications

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Abstract. After shortly reviewing the fundamentals of approach theory as introduced by R. Lowen in 1989, we show that this theory is intimately related with the well-known Wasserstein metric on the space of probability measures with a finite first moment on a complete and separable metric space. More precisely, we introduce a canonical approach structure, called the contractive approach structure, and prove that it is metrized by the Wasserstein metric. The key ingredients of the proof of this result are Dini's Theorem, Ascoli's Theorem, and the fact that the class of real-valued contractions on a metric space has some nice stability properties. We then combine the obtained result with Prokhorov's Theorem to establish inequalities between the relative Hausdorff measure of non-compactness for the Wasserstein metric and a canonical measure of non-uniform integrability.

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“…But the dual of a quantified space has a richer structure: the homset KX := [X, R] of a locally convex approach space X is the closed unit ball of a seminorm on the space of all linear continuous functionals on X; the resulting seminormed space is denoted L b K X [15]. There is also a converse connection ( [15], 2.11): if we start with a vector space X and closed unit ball X in the algebraic dual of X and endow X with M (X,X ) , the inital lcApVec structure of the source (X…”

Section: Introductionmentioning

confidence: 99%

Every Banach Space is Reflexive

Olmen

Verwulgen

2006

Appl Categor Struct

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The title above is wrong, because the strong dual of a Banach space is too strong to assert that the natural correspondence between a space and its bidual is an isomorphism. This, from a categorical point of view, is indeed the right duality concept because it yields a self adjoint dualisation functor. However, for many applications the non-reflexiveness problem can be solved by replacing the norm on the first dual by the weak*-structure [1].But then, by taking the second dual, only the original vector space is recovered and no universal property remains with this modified dual structure. In this work we unify the applied and the structural point of view.We introduce a suitable numerical structure on vector spaces such that Banach balls, or more precisely totally convex modules, arise naturally in duality, i.e. as a category of Eilenberg-Moore algebras. This numerical structure naturally overlies the weak*-topology on the algebraic dual, so the entire Banach space can be reconstructed as a second dual. Moreover, the isomorphism between the original space and its bidual is the unit of an adjunction between the two dualisation functors.Notice that the weak*-topology is normable only if it lives on a finite dimensional space; in that case the original space is trivial as well, hence reflexive. So the overlying numerical structure should be something more general than a norm or a seminorm and thus approach theory [2,3] enters the picture.

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Quantified functional analysis: recapturing the dual unit ball

Cited by 5 publications

References 11 publications

An Isometric Representation of the Dual of $ {\user1{\mathcal{C}}}{\left( {X,\mathbb{R}} \right)} $

An Isometric Representation of the Dual of $ {\user1{\mathcal{C}}}{\left( {X,\mathbb{R}} \right)} $

An application of approach theory to the relative Hausdorff measure of non-compactness for the Wasserstein metric

Every Banach Space is Reflexive

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