scholarly journals New Results on the Aggregation of Norms

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2291
Author(s):  
Tatiana Pedraza ◽  
Jesús Rodríguez-López
Keyword(s):  
Vector Space ◽  
Metric Spaces ◽  
Cartesian Product ◽  
Topological Spaces ◽  
Factor Space ◽  
Natural Question ◽  
Product Topology ◽  
Aggregation Procedure ◽  
Arbitrary Collection

It is a natural question if a Cartesian product of objects produces an object of the same type. For example, it is well known that a countable Cartesian product of metrizable topological spaces is metrizable. Related to this question, Borsík and Doboš characterized those functions that allow obtaining a metric in the Cartesian product of metric spaces by means of the aggregation of the metrics of each factor space. This question was also studied for norms by Herburt and Moszyńska. This aggregation procedure can be modified in order to construct a metric or a norm on a certain set by means of a family of metrics or norms, respectively. In this paper, we characterize the functions that allow merging an arbitrary collection of (asymmetric) norms defined over a vector space into a single norm (aggregation on sets). We see that these functions are different from those that allow the construction of a norm in a Cartesian product (aggregation on products). Moreover, we study a related topological problem that was considered in the context of metric spaces by Borsík and Doboš. Concretely, we analyze under which conditions the aggregated norm is compatible with the product topology or the supremum topology in each case.

Topologies Extending Valuations

1978 ◽  
Vol 30 (01) ◽  
pp. 164-169 ◽  
Author(s):  
Thomas Rigo ◽  
Seth Warner
Keyword(s):  
Vector Space ◽  
Cartesian Product ◽  
Field Extension ◽  
Valuation Theory ◽  
Topological Isomorphism ◽  
Product Topology ◽  
Hausdorff Topology ◽  
Absolute Value ◽  
Finite Dimensional ◽  
Multilinear Mapping

Let K be a field complete for a proper valuation (absolute value) v. It is classic that a finite-dimensional K-vector space E admits a unique Hausdorff topology making it a topological K-vector space, and that that topology is the “cartesian product topology” in the sense that for any basis c1 …, cn of E, is a topological isomorphism from K n to E [1, Chap. I, § 2, no. 3; 2, Chap. VI, § 5, no. 2]. It follows readily that any multilinear mapping from E m to a Hausdorff topological K-vector space is continuous. In particular, any multiplication on E making it a K-algebra is continuous in both variables. If for some such multiplication E is a field extension of K, then by valuation theory the unique Hausdorff topology of E is given by a valuation (absolute value) extending v.

scholarly journals Caristi Type Coincidence Point Theorem in Topological Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Jiang Zhu ◽  
Lei Wei ◽  
Cheng-Cheng Zhu
Keyword(s):  
Variational Principle ◽  
Vector Space ◽  
Point Theorem ◽  
Metric Spaces ◽  
Coincidence Point ◽  
Topological Spaces ◽  
Uniform Spaces ◽  
Type Function ◽  
Fuzzy Metric Spaces ◽  
Semimetric Spaces

A generalized Caristi type coincidence point theorem and its equivalences in the setting of topological spaces by using a kind of nonmetric type function are obtained. These results are used to establish variational principle and its equivalences ind-complete spaces, bornological vector space, seven kinds of completed quasi-semimetric spaces equipped withQ-functions, uniform spaces withq-distance, generating spaces of quasimetric family, and fuzzy metric spaces.

scholarly journals A contribution to the study of fuzzy metric spaces

2001 ◽  
Vol 2 (1) ◽  
pp. 63 ◽  
Author(s):  
Almanzor Sapena Piera
Keyword(s):  
Metric Spaces ◽  
Fuzzy Theory ◽  
Topological Spaces ◽  
Fuzzy Metric Spaces ◽  
Fuzzy Metric

We give some examples nad properties of fuzzy metric spaces, in the sense of George and Veramani, and characterize the To topological spaces which admit a compatible uniformity that has a countable transitive base, in terms of the fuzzy theory.

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.

Characterization of Certain Classes of Spaces With Gδ Points as Open Images of Metric Spaces

1975 ◽  
Vol 27 (6) ◽  
pp. 1229-1238
Author(s):  
Kenneth C. Abernethy
Keyword(s):  
Metric Spaces ◽  
Topological Spaces ◽  
Generalized Metric Spaces ◽  
The Past ◽  
Generalized Metric

The study of metrization has led to the development of a number of new topological spaces, called generalized metric spaces, within the past fifteen years. For a survey of results in metrization theory involving many of these spaces, the reader is referred to [13]. Quite a few of these generalized metric spaces have been studied extensively, somewhat independently of their role in metrization theorems. Specifically, we refer here to characterizations of these spaces by various workers as images of metric spaces. Results in this area have been obtained by Alexander [2], Arhangel'skii [3], Burke [5], Heath [10], Michael [15], Nagata [16], and the author [1], to mention a few. Later we will recall specifically some of these results.

On the Vanishing of a (G, σ) Product in a (G, σ) Space

1970 ◽  
Vol 22 (2) ◽  
pp. 363-371 ◽  
Author(s):  
K. Singh
Keyword(s):  
Vector Space ◽  
Multiplicative Group ◽  
Cartesian Product ◽  
Arbitrary Field ◽  
Necessary And Sufficient Condition ◽  
Tensor Space ◽  
Linear Character ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Necessary And Sufficient

In this paper, we shall construct a vector space, called the (G, σ) space, which generalizes the tensor space, the Grassman space, and the symmetric space. Then we shall determine a necessary and sufficient condition that the (G, σ) product of the vectors x1, x2, …, xn is zero.1. Let G be a permutation group on I = {1, 2, …, n} and F, an arbitrary field. Let σ be a linear character of G, i.e., σ is a homomorphism of G into the multiplicative group F* of F.For each i ∈ I, let Vi be a finite-dimensional vector space over F. Consider the Cartesian product W = V1 × V2 × … × Vn.1.1. Definition. W is called a G-set if and only if Vi = Vg(i) for all i ∊ I, and for all g ∊ G.

scholarly journals Monad Metrizable Space

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1891
Author(s):  
Orhan Göçür
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Real Space ◽  
Metrizable Space ◽  
Topological Spaces ◽  
Metrizable Spaces

Do the topologies of each dimension have to be same and metrizable for metricization of any space? I show that this is not necessary with monad metrizable spaces. For example, a monad metrizable space may have got any indiscrete topologies, discrete topologies, different metric spaces, or any topological spaces in each different dimension. I compute the distance in real space between such topologies. First, the passing points between different topologies is defined and then a monad metric is defined. Then I provide definitions and some properties about monad metrizable spaces and PAS metric spaces. I show that any PAS metric space is also a monad metrizable space. Moreover, some properties and some examples about them are presented.

scholarly journals Isometries of spaces of unbounded continuous functions

2001 ◽  
Vol 63 (3) ◽  
pp. 475-484
Author(s):  
Jesús Araujo ◽  
Krzysztof Jarosz
Keyword(s):  
Banach Spaces ◽  
Metric Spaces ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Compact Sets ◽  
Surjective Isometry ◽  
Bounded Continuous Functions ◽  
Stone Theorem

By the classical Banach-Stone Theorem any surjective isometry between Banach spaces of bounded continuous functions defined on compact sets is given by a homeomorphism of the domains. We prove that the same description applies to isometries of metric spaces of unbounded continuous functions defined on non compact topological spaces.

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