scholarly journals Topological Properties of Real Normed Space

2014 ◽  
Vol 22 (3) ◽  
pp. 209-223
Author(s):  
Kazuhisa Nakasho ◽  
Yuichi Futa ◽  
Yasunari Shidama
Keyword(s):  
Vector Space ◽  
Normed Space ◽  
Lipschitz Continuity ◽  
Inverse Image ◽  
Linear Functions ◽  
Topological Properties ◽  
Normed Spaces ◽  
Real Normed Space ◽  
Convergence Of Sequences ◽  
Normed Subspace

Summary In this article, we formalize topological properties of real normed spaces. In the first part, open and closed, density, separability and sequence and its convergence are discussed. Then we argue properties of real normed subspace. Then we discuss linear functions between real normed speces. Several kinds of subspaces induced by linear functions such as kernel, image and inverse image are considered here. The fact that Lipschitz continuity operators preserve convergence of sequences is also refered here. Then we argue the condition when real normed subspaces become Banach’s spaces. We also formalize quotient vector space. In the last session, we argue the properties of the closure of real normed space. These formalizations are based on [19](p.3-41), [2] and [34](p.3-67).

scholarly journals Bidual Spaces and Reflexivity of Real Normed Spaces

2014 ◽  
Vol 22 (4) ◽  
pp. 303-311
Author(s):  
Keiko Narita ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Real Number ◽  
Normed Space ◽  
Uniform Boundedness ◽  
Linear Operators ◽  
Linear Functionals ◽  
Normed Spaces ◽  
Boundedness Theorem ◽  
Natural Mapping ◽  
Real Normed Space ◽  
Banach Theorem

Summary In this article, we considered bidual spaces and reflexivity of real normed spaces. At first we proved some corollaries applying Hahn-Banach theorem and showed related theorems. In the second section, we proved the norm of dual spaces and defined the natural mapping, from real normed spaces to bidual spaces. We also proved some properties of this mapping. Next, we defined real normed space of R, real number spaces as real normed spaces and proved related theorems. We can regard linear functionals as linear operators by this definition. Accordingly we proved Uniform Boundedness Theorem for linear functionals using the theorem (5) from [21]. Finally, we defined reflexivity of real normed spaces and proved some theorems about isomorphism of linear operators. Using them, we proved some properties about reflexivity. These formalizations are based on [19], [20], [8] and [1].

scholarly journals Some remarks on functions with values in probabilistic normed spaces

2007 ◽  
Vol 57 (3) ◽  
Author(s):  
Ioan Goleţ
Keyword(s):  
Normed Space ◽  
Topological Properties ◽  
Normed Spaces ◽  
Probabilistic Normed Space ◽  
Random Functions ◽  
New Class ◽  
Probabilistic Normed Spaces ◽  
Special Cases ◽  
Probabilistic Norm

AbstractIn this paper we consider an enlargement of the notion of the probabilistic normed space. For this new class of probabilistic normed spaces we give some topological properties. By using properties of the probabilistic norm we prove some differential and integral properties of functions with values into probabilistic normed spaces. As special cases, results for deterministic and random functions can be obtained.

scholarly journals Ideally Statistical Convergence in n-normed Space

2020 ◽  
pp. 75-84
Author(s):  
Nazneen Khan ◽  
Amani Shatarah
Keyword(s):  
Normed Space ◽  
Cauchy Sequence ◽  
Statistical Convergence ◽  
Topological Properties ◽  
Normed Spaces

The aim of the article is to extend the concept of Ideally statistical convergence from 2 normed spaces to n-normed space. We have also study and prove some important algebraic and topological properties of Ideally-statistical convergence of real sequences in n-normed space. In the last part of this article we obtain a criterion for I-statistically Cauchy sequence in n-normed space to be I-statistically Cauchy with respect to ∥.∥∞.

Generalized Sequence Spaces in 2-Normed Spaces Defined by Ideal and a Modulus Function

2014 ◽  
Vol 0 (0) ◽  
Author(s):  
Sudhir Kumar ◽  
Vijay Kumar ◽  
S.S. Bhatia
Keyword(s):  
Normed Space ◽  
Difference Operator ◽  
Convergent Sequence ◽  
Sequence Spaces ◽  
Modulus Function ◽  
Topological Properties ◽  
Normed Spaces ◽  
New Class ◽  
Difference Sequences ◽  
Convergent Sequences

AbstractThe main objective of this paper is to define some new kind of generalized convergent sequence spaces with respect to a modulus function, and difference operator Δm, m ≥ 1 in a 2-normed space. We also examine some topological properties of the resulting sequence spaces. Finally, we have introduced a new class of generalized convergent sequences with the help of an ideal and difference sequences in the same space.

scholarly journals Maximum average distance in complex finite dimensional normed spaces

2002 ◽  
Vol 66 (1) ◽  
pp. 125-134
Author(s):  
Juan C. García-Vázquez ◽  
Rafael Villa
Keyword(s):  
Unit Sphere ◽  
Metric Space ◽  
Normed Space ◽  
Average Distance ◽  
Complex Case ◽  
Normed Spaces ◽  
Finite Dimensional ◽  
Real Normed Space ◽  
Dimensional Normed Space

A number r > 0 is called a rendezvous number for a metric space (M, d) if for any n ∈ ℕ and any x1,…xn ∈ M, there exists x ∈ M such that . A rendezvous number for a normed space X is a rendezvous number for its unit sphere. A surprising theorem due to O. Gross states that every finite dimensional normed space has one and only one average number, denoted by r (X). In a recent paper, A. Hinrichs solves a conjecture raised by R. Wolf. He proves that for any n-dimensional real normed space. In this paper, we prove the analogous inequality in the complex case for n ≥ 3.

scholarly journals Classes ofI-Convergent Double Sequences overn-Normed Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
Nazneen Khan
Keyword(s):  
Normed Space ◽  
Topological Properties ◽  
Double Sequences ◽  
Normed Spaces ◽  
Inclusion Relations

I introduce some new classes ofI-convergent double sequences defined by a sequence of moduli overn-normed space. Study of their algebraic and topological properties and some inclusion relations has also been done.

scholarly journals On normed spaces with the Wigner Property

2019 ◽  
Vol 11 (3) ◽  
pp. 523-539 ◽  
Author(s):  
Ruidong Wang ◽  
Dariusz Bugajewski
Keyword(s):  
Normed Space ◽  
Two Dimensional ◽  
Normed Spaces ◽  
Strictly Convex ◽  
Real Normed Space

AbstractThe aim of this paper is to generalize the Wigner Theorem to real normed spaces. A normed space is said to have the Wigner Property if the Wigner Theorem holds for it. We prove that every two-dimensional real normed space has the Wigner Property. We also study the Wigner Property of real normed spaces of dimension at least three. It is also shown that strictly convex real normed spaces possess the Wigner Property.

scholarly journals Riemann Integral of Functions from ℝ into Real Banach Space

2013 ◽  
Vol 21 (2) ◽  
pp. 145-152
Author(s):  
Keiko Narita ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Banach Space ◽  
Normed Space ◽  
Real Banach Space ◽  
Closed Interval ◽  
Continuous Functions ◽  
Uniform Continuity ◽  
Riemann Integral ◽  
Real Normed Space ◽  
Finite Sequences ◽  
Convergence Of Sequences

Summary In this article we deal with the Riemann integral of functions from R into a real Banach space. The last theorem establishes the integrability of continuous functions on the closed interval of reals. To prove the integrability we defined uniform continuity for functions from R into a real normed space, and proved related theorems. We also stated some properties of finite sequences of elements of a real normed space and finite sequences of real numbers. In addition we proved some theorems about the convergence of sequences. We applied definitions introduced in the previous article [21] to the proof of integrability.

scholarly journals Remarks on the Conditional Quadratic and the Conditional D’Alembert Functional Equations on Particular Normed Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Margherita Fochi ◽  
Gabriella Viola
Keyword(s):  
Functional Equation ◽  
Normed Space ◽  
Functional Equations ◽  
Quadratic Equation ◽  
Restricted Domain ◽  
Normed Spaces ◽  
Cauchy Functional Equation ◽  
Real Normed Space

Let be a real normed space with dimension greater than 2 and let be a real functional defined on . Applying some ideas from the studies made on the conditional Cauchy functional equation on the restricted domain of the vectors of equal norm and the isosceles orthogonal vectors, the conditional quadratic equation and the D’Alembert one, namely, and , have been studied in this paper, in order to describe their solutions. Particular normed spaces are introduced for this aim.

scholarly journals ON SOME TOPOLOGICAL PROPERTIES IN GRADUAL NORMED SPACES

2020 ◽  
pp. 549
Author(s):  
Mina Ettefagh ◽  
Farnaz Y. Azari ◽  
Sina Etemad
Keyword(s):  
Normed Space ◽  
Topological Properties ◽  
Compact Sets ◽  
Normed Spaces ◽  
Closed Sets ◽  
The Relationship

In this paper, we have investigated some topological properties of sets in a given gradual normed space. We have stated gradual Hausdorff property and then,we have studied the relationship between gradual closed sets and gradual compact sets. Also, we have given a result about having the closure point for an innite set in a gradual normed space. In the end, we have provided some illustrative examples.

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