scholarly journals Internal functionals and bundle duals

1984 ◽  
Vol 7 (4) ◽  
pp. 689-695 ◽  
Author(s):  
Joseph W. Kitchen ◽  
David A. Robbins
Keyword(s):  
Banach Spaces ◽  
Vector Spaces ◽  
Topological Spaces ◽  
Topological Vector Spaces ◽  
The Given ◽  
Locally Convex

Ifπ:E→Xis a bundle of Banach spaces,Xcompact Hausdorff, a fibered spaceπ*:E*→Xcan be constructed whose stalks are the duals of the stalks of the given bundle and whose sections can be identified with the “functionals” studied by Seda in [1] and [2] or elements of the “internal dual”Mod(Γ(π),C(X))studied by Gierz in [3]. If the given bundle is separable and norm continuous, then the fibered spaceπ*:E*→Xis actually a full bundle of locally convex topological vector spaces (Theorem 3). In the second portion of the paper two results are stated, both of them corollaries of theorems by Gierz, concerning functionals for bundles of Banach spaces which arise, in turn, from “fields of topological spaces.”

An Embedding Theorem for Separable Locally Convex Spaces

1971 ◽  
Vol 14 (1) ◽  
pp. 119-120 ◽  
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.

scholarly journals Bornologiese pseudotopologiese vektorruimtes

1990 ◽  
Vol 9 (1) ◽  
pp. 15-18
Author(s):  
M. A. Muller
Keyword(s):  
Vector Space ◽  
Inductive Limit ◽  
Vector Spaces ◽  
Topological Spaces ◽  
Direct Sums ◽  
Paper Properties ◽  
Topological Vector Spaces ◽  
Quotient Spaces ◽  
Locally Convex

Homological spaces were defined by Hogbe-Nlend in 1971 and pseudo-topological spaces by Fischer in 1959. In this paper properties of bornological pseudo-topological vector spaces are investigated. A characterization of such spaces is obtained and it is shown that quotient spaces and direct sums o f boruological pseudo-topological vector spaces are bornological. Every bornological locally convex pseudo-topological vector space is shown to be the inductive limit in the category of pseudo-topological vector spaces of a family of locally convex topological vector spaces.

scholarly journals S-Barrelled Topological Vector Spaces*

1978 ◽  
Vol 21 (2) ◽  
pp. 221-227 ◽  
Author(s):  
Ray F. Snipes
Keyword(s):  
Banach Spaces ◽  
Vector Spaces ◽  
Fréchet Spaces ◽  
Topological Vector Spaces ◽  
Barrelled Spaces ◽  
Steinhaus Theorem ◽  
Locally Convex

N. Bourbaki [1] was the first to introduce the class of locally convex topological vector spaces called “espaces tonnelés” or “barrelled spaces.” These spaces have some of the important properties of Banach spaces and Fréchet spaces. Indeed, a generalized Banach-Steinhaus theorem is valid for them, although barrelled spaces are not necessarily metrizable. Extensive accounts of the properties of barrelled locally convex topological vector spaces are found in [5] and [8].

Open decompositions on ordered convex spaces

1973 ◽  
Vol 74 (1) ◽  
pp. 49-59 ◽  
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?

scholarly journals On The Closed Graph Theorem

1956 ◽  
Vol 3 (1) ◽  
pp. 9-12 ◽  
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.

Finite dimensional locally convex cones

Filomat ◽  
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.

scholarly journals ULTRAMETRIC AND NON-LOCALLY CONVEX ANALOGUES OF THE GENERAL CURVE LEMMA OF CONVENIENT DIFFERENTIAL CALCULUS

2008 ◽  
Vol 50 (2) ◽  
pp. 271-288
Author(s):  
HELGE GLÖCKNER
Keyword(s):  
Differential Calculus ◽  
Vector Spaces ◽  
Local Convexity ◽  
Topological Vector Spaces ◽  
Infinite Dimensional ◽  
General Curve ◽  
Infinite Dimensional Analysis ◽  
Locally Convex ◽  
Differentiability Properties ◽  
Made In

AbstractThe General Curve Lemma is a tool of Infinite-Dimensional Analysis that enables refined studies of differentiability properties of maps between real locally convex spaces to be made. In this article, we generalize the General Curve Lemma in two ways. First, we remove the condition of local convexity in the real case. Second, we adapt the lemma to the case of curves in topological vector spaces over ultrametric fields.

scholarly journals Newel-pseudotopologiese vektorruimtes

1999 ◽  
Vol 18 (3) ◽  
pp. 89-93
Author(s):  
M. A. Muller
Keyword(s):  
Vector Spaces ◽  
Topological Spaces ◽  
Topological Vector Spaces

Pseudo-topological spaces (i.e. limit spaces) were defined by Fischer in 1959. In this paper the theory of fuzzy pseudo-topological spaces is applied to vector spaces. We introduce the concept of boundedness in fuzzy pseudo-topological vector spaces.

scholarly journals M – STAR – Irresolute Topological Vector Spaces

2021 ◽  
Vol 7 ◽  
pp. 20-36
Author(s):  
Raja Mohammad Latif
Keyword(s):  
Vector Space ◽  
Extreme Point ◽  
Topological Vector Space ◽  
Convex Subset ◽  
Hausdorff Space ◽  
Vector Spaces ◽  
Topological Spaces ◽  
Topological Vector Spaces ◽  
New Class

In 2016 A. Devika and A. Thilagavathi introduced a new class of sets called M*-open sets and investigated some properties of these sets in topological spaces. In this paper, we introduce and study a new class of spaces, namely M*-irresolute topological vector spaces via M*-open sets. We explore and investigate several properties and characterizations of this new notion of M*-irresolute topological vector space. We give several characterizations of M*-Hausdorff space. Moreover, we show that the extreme point of the convex subset of M*-irresolute topological vector space X lies on the boundary.

scholarly journals Linear mappings between topological vector spaces

1972 ◽  
Vol 14 (1) ◽  
pp. 105-118
Author(s):  
B. D. Craven
Keyword(s):  
Inductive Limit ◽  
Vector Spaces ◽  
Continuous Linear ◽  
Additional Structure ◽  
Normed Spaces ◽  
Topological Vector Spaces ◽  
Locally Convex

If A and B are locally convex topological vector spaces, and B has certain additional structure, then the space L(A, B) of all continuous linear mappings of A into B is characterized, within isomorphism, as the inductive limit of a family of spaces, whose elements are functions, or measures. The isomorphism is topological if L(A, B) is given a particular topology, defined in terms of the seminorms which define the topologies of A and B. The additional structure on B enables L(A, B) to be constructed, using the duals of the normed spaces obtained by giving A the topology of each of its seminorms separately.

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