scholarly journals On barrelled topological modules

1996 ◽  
Vol 19 (1) ◽  
pp. 45-52 ◽  
Author(s):  
Dinamérico P. Pombo
Keyword(s):  
Vector Spaces ◽  
Topological Vector Spaces ◽  
Topological Modules

Some of the main theorems concerning ultrabarrelled topological vector spaces are extended to the context of topological modules.

Geometry of balanced and absorbing subsets of topological modules

2019 ◽  
Vol 18 (06) ◽  
pp. 1950119 ◽  
Author(s):  
Francisco Javier García-Pacheco ◽  
Pablo Piniella
Keyword(s):  
Unit Ball ◽  
Extreme Points ◽  
Vector Spaces ◽  
Closed Unit Ball ◽  
Absorbing Set ◽  
Internal Point ◽  
Topological Vector Spaces ◽  
Topological Module ◽  
Real Algebra ◽  
Topological Modules

We define the concepts of balanced set and absorbing set in modules over topological rings, which coincide with the usual concepts when restricting to topological vector spaces. We show that in a topological module over an absolute semi-valued ring whose invertibles approach [Formula: see text], every neighborhood of [Formula: see text] is absorbing. We also introduce the concept of total closed unit neighborhood of zero (total closed unit) and prove that the only total closed unit of the quaternions [Formula: see text] is its closed unit ball [Formula: see text]. On the other hand, we also prove that if [Formula: see text] is an absolute semi-valued unital real algebra, then its closed unit ball [Formula: see text] is a total closed unit. Finally, we study the geometry of modules via the extreme points and the internal points, showing that no internal point can be an extreme point and that absorbance is equivalent to having [Formula: see text] as an internal point.

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 *—Inductive limits and partition of unity

1989 ◽  
Vol 12 (3) ◽  
pp. 429-434
Author(s):  
V. Murali
Keyword(s):  
Inductive Limit ◽  
Partition Of Unity ◽  
Vector Spaces ◽  
Inductive Limits ◽  
Topological Vector Spaces ◽  
Limit Space ◽  
Direct Sum

In this note we define and discuss some properties of partition of unity on *-inductive limits of topological vector spaces. We prove that if a partition of unity exists on a *-inductive limit space of a collection of topological vector spaces, then it is isomorphic and homeomorphic to a subspace of a *-direct sum of topological vector spaces.

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