Geometry of balanced and absorbing subsets of topological modules

2019 ◽  
Vol 18 (06) ◽  
pp. 1950119 ◽  
Author(s):  
Francisco Javier García-Pacheco ◽  
Pablo Piniella

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.

2003 ◽  
Vol 67 (3) ◽  
pp. 365-375
Author(s):  
K. J. Wirths ◽  
J. Xiao

For p ∈ (0, ∞) define Qp0(∂Δ) as the space of all Lebesgue measurable complex-valued functions f; on the unit circle ∂Δ for which ∫∂Δf;(z)|dz|/(2π) = 0 andas the open subarc I of ∂Δ varies. Note that each Qp,0(∂Δ) lies between the Dirichlet space and Sarason's vanishing mean oscillation space. This paper determines the extreme points of the closed unit ball of Qp,0(∂Δ) equipped with an appropriate norm.


2020 ◽  
Vol 27 (3) ◽  
pp. 361-366 ◽  
Author(s):  
Francisco Javier García-Pacheco ◽  
Enrique Naranjo-Guerra

AbstractInternal points were introduced in the literature of topological vector spaces to characterize the finest locally convex vector topology. In this manuscript we generalize the concept of internal point in real vector spaces by introducing a type of points, called inner points, that allows us to provide an intrinsic characterization of linear manifolds, which was not possible by using internal points. We also characterize infinite dimensional real vector spaces by means of the inner points of convex sets. Finally, we prove that in convex sets containing internal points, the set of inner points coincides with the one of internal points.


Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1580
Author(s):  
Francisco Javier Garcia-Pacheco

The framework of Functional Analysis is the theory of topological vector spaces over the real or complex field. The natural generalization of these objects are the topological modules over topological rings. Weakening the classical Functional Analysis results towards the scope of topological modules is a relatively new trend that has enriched the literature of Functional Analysis with deeper classical results as well as with pathological phenomena. Following this trend, it has been recently proved that every real or complex Hausdorff locally convex topological vector space with dimension greater than or equal to 2 has a balanced and absorbing subset with empty interior. Here we propose an extension of this result to topological modules over topological rings. A sufficient condition is provided to accomplish this extension. This sufficient condition is a new property in topological module theory called strong open property. On the other hand, topological regularity of closed balls and open balls in real or complex normed spaces is a trivial fact. Sufficient conditions, related to the strong open property, are provided on seminormed modules over an absolutely semivalued ring for closed balls to be regular closed and open balls to be regular open. These sufficient conditions are in fact characterizations when the seminormed module is the absolutely semivalued ring. These characterizations allow the provision of more examples of closed-unit neighborhoods of zero. Consequently, the closed-unit ball of any unital real Banach algebra is proved to be a closed-unit zero-neighborhood. We finally transport all these results to topological modules over topological rings to obtain nontrivial regular closed and regular open neighborhoods of zero. In particular, if M is a topological R-module and m∗∈M∗ is a continuous linear functional on M which is open as a map between topological spaces, then m∗−1(int(B)) is regular open and m∗−1(B) is regular closed, for B any closed-unit zero-neighborhood in R.


Author(s):  
Dinamérico P. Pombo

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


2016 ◽  
Vol 95 (2) ◽  
pp. 315-321
Author(s):  
ANA M. CABRERA-SERRANO ◽  
JUAN F. MENA-JURADO

We say that a Banach space $X$ is ‘nice’ if every extreme operator from any Banach space into $X$ is a nice operator (that is, its adjoint preserves extreme points). We prove that if $X$ is a nice almost $CL$-space, then $X$ is isometrically isomorphic to $c_{0}(I)$ for some set $I$. We also show that if $X$ is a nice Banach space whose closed unit ball has the Krein–Milman property, then $X$ is $l_{\infty }^{n}$ for some $n\in \mathbb{N}$. The proof of our results relies on the structure topology.


Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5111-5116
Author(s):  
Davood Ayaseha

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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