Predicate transformers for extended probability and non-determinism

We investigate laws for predicate transformers for the combination of non-deterministic choice and (extended) probabilistic choice, where predicates are taken to be functions to the extended non-negative reals, or to closed intervals of such reals. These predicate transformers correspond to state transformers, which are functions to conical powerdomains, which are the appropriate powerdomains for the combined forms of non-determinism. As with standard powerdomains for non-deterministic choice, these come in three flavours – lower, upper and (order-)convex – so there are also three kinds of predicate transformers. In order to make the connection, the powerdomains are first characterised in terms of relevant classes of functionals.Much of the development is carried out at an abstract level, a kind of domain-theoretic functional analysis: one considers d-cones, which are dcpos equipped with a module structure over the non-negative extended reals, in place of topological vector spaces. Such a development still needs to be carried out for probabilistic choice per se; it would presumably be necessary to work with a notion of convex space rather than a cone.

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On The Closed Graph Theorem

Proceedings of the Glasgow Mathematical Association ◽

10.1017/s2040618500033372 ◽

1956 ◽

Vol 3 (1) ◽

pp. 9-12 ◽

Cited By ~ 33

Author(s):

Alex. P. Robertson ◽

Wendy Robertson

Keyword(s):

Functional Analysis ◽

Banach Spaces ◽

Convex Space ◽

Vector Spaces ◽

Complex Field ◽

Closed Graph ◽

Graph Theorem ◽

Closed Graph Theorem ◽

Topological Vector Spaces ◽

Locally Convex

The closed graph theorem is one of the deeper results in the theory of Banach spaces and one of the richest in its applications to functional analysis. This note contains an extension of the theorem to certain classes of topological vector spaces. For the most part, we use the terminology and notation of N. Bourbaki [1], contracting “locally convex topological vector space over the real or complex field” to “convex space”; here we confine ourselves to convex spaces.

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An Embedding Theorem for Separable Locally Convex Spaces

Canadian Mathematical Bulletin ◽

10.4153/cmb-1971-023-1 ◽

1971 ◽

Vol 14 (1) ◽

pp. 119-120 ◽

Cited By ~ 1

Author(s):

Robert H. Lohman

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Convex Space ◽

Separable Banach Space ◽

Locally Convex Space ◽

Embedding Theorem ◽

Vector Spaces ◽

Locally Convex Spaces ◽

Topological Vector Spaces ◽

Locally Convex

A well-known embedding theorem of Banach and Mazur [1, p. 185] states that every separable Banach space is isometrically isomorphic to a subspace of C[0, 1], establishing C[0, 1] as a universal separable Banach space. The embedding theorem one encounters in a course in topological vector spaces states that every Hausdorff locally convex space (l.c.s.) is topologically isomorphic to a subspace of a product of Banach spaces.

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Topological Quasilinear Spaces

Abstract and Applied Analysis ◽

10.1155/2012/951374 ◽

2012 ◽

Vol 2012 ◽

pp. 1-10 ◽

Cited By ~ 4

Author(s):

Yılmaz Yılmaz ◽

Sümeyye Çakan ◽

Şahika Aytekin

Keyword(s):

Functional Analysis ◽

Linear Functional ◽

Vector Spaces ◽

Multivalued Mappings ◽

Linear Spaces ◽

Localization Principle ◽

Topological Vector Spaces

We introduce, in this work, the notion of topological quasilinear spaces as a generalization of the notion of normed quasilinear spaces defined by Aseev (1986). He introduced a kind of the concept of a quasilinear spaces both including a classical linear spaces and also nonlinear spaces of subsets and multivalued mappings. Further, Aseev presented some basic quasilinear counterpart of linear functional analysis by introducing the notions of norm and bounded quasilinear operators and functionals. Our investigations show that translation may destroy the property of being a neighborhood of a set in topological quasilinear spaces in contrast to the situation in topological vector spaces. Thus, we prove that any topological quasilinear space may not satisfy the localization principle of topological vector spaces.

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Some properties of vector measures taking values in a topological vector space

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700029360 ◽

1987 ◽

Vol 43 (2) ◽

pp. 224-230

Author(s):

Efstathios Giannakoulias

Keyword(s):

Vector Space ◽

Topological Vector Space ◽

Convex Space ◽

Locally Convex Space ◽

Vector Spaces ◽

Necessary Condition ◽

Vector Measures ◽

Topological Vector Spaces ◽

Locally Convex ◽

Pettis Integrability

AbstractIn this paper we study some properties of vector measures with values in various topological vector spaces. As a matter of fact, we give a necessary condition implying the Pettis integrability of a function f: S → E, where S is a set and E a locally convex space. Furthermore, we prove an iff condition under which (Q, E) has the Pettis property, for an algebra Q and a sequentially complete topological vector space E. An approximating theorem concerning vector measures taking values in a Fréchet space is also given.

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Open decompositions on ordered convex spaces

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100047770 ◽

1973 ◽

Vol 74 (1) ◽

pp. 49-59 ◽

Cited By ~ 3

Author(s):

Yau-Chuen Wong

Keyword(s):

Vector Space ◽

Banach Spaces ◽

Convex Space ◽

Locally Convex Space ◽

Positive Cone ◽

Vector Spaces ◽

Topological Vector Spaces ◽

Closed Cone ◽

Necessary And Sufficient ◽

Locally Convex

Let (E, ) be a topological vector space with a positive cone C. Jameson (3) says that C given an open decomposition on E if V ∩ C − V ∩ C is a -neighbourhood of 0 whenever V is a -neighbourhood of 0. The concept of open decompositions plays an important rôle in the theory of ordered topological vector spaces; see (3). It is clear that C is generating if C gives an open decomposition on E; the converse is true for Banach spaces with a closed cone, by Andô's theorem (cf. (1) or (9)). Therefore the following question arises naturally:(Q 1) Let (E, ) be a locally convex space with a positive cone C. What condition on is necessary and sufficient for the cone C to give an open decomposition on E?

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ON SOFT FUZZY SOFT TOPOLOGICAL VECTOR SPACES AND DIFFERENTIATIONS

Advances in Mathematics: Scientific Journal ◽

10.37418/amsj.9.8.82 ◽

2020 ◽

Vol 9 (8) ◽

pp. 6145-6151

Author(s):

V. Visalakshi ◽

T. Yogalakshmi

Keyword(s):

Vector Spaces ◽

Topological Vector Spaces

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Finite dimensional locally convex cones

Filomat ◽

10.2298/fil1716111a ◽

2017 ◽

Vol 31 (16) ◽

pp. 5111-5116

Author(s):

Davood Ayaseha

Keyword(s):

Finite Dimension ◽

Convex Cones ◽

Vector Spaces ◽

Topological Vector Spaces ◽

Euclidean Topology ◽

Finite Dimensional ◽

Dimensional Cone ◽

Locally Convex Cones ◽

Locally Convex ◽

Special Case

We study the locally convex cones which have finite dimension. We introduce the Euclidean convex quasiuniform structure on a finite dimensional cone. In special case of finite dimensional locally convex topological vector spaces, the symmetric topology induced by the Euclidean convex quasiuniform structure reduces to the known concept of Euclidean topology. We prove that the dual of a finite dimensional cone endowed with the Euclidean convex quasiuniform structure is identical with it?s algebraic dual.

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