Ideals of Operators on Complete Inner Product Spaces and on Banach Spaces

1987 ◽  
pp. 394-426
Author(s):  
Vasile Ion Istrăţescu
Keyword(s):  
Banach Spaces ◽  
Inner Product ◽  
Product Spaces ◽  
Inner Product Spaces

scholarly journals REVISITING THE RECTANGULAR CONSTANT IN BANACH SPACES

2021 ◽  
pp. 1-10
Author(s):  
M. BARONTI ◽  
E. CASINI ◽  
P. L. PAPINI
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Real Banach Space ◽  
Inner Product ◽  
Two Dimensional ◽  
Product Spaces ◽  
Inner Product Spaces

Abstract Let X be a real Banach space. The rectangular constant $\mu (X)$ and some generalisations of it, $\mu _p(X)$ for $p \geq 1$ , were introduced by Gastinel and Joly around half a century ago. In this paper we make precise some characterisations of inner product spaces by using $\mu _p(X)$ , correcting some statements appearing in the literature, and extend to $\mu _p(X)$ some characterisations of uniformly nonsquare spaces, known only for $\mu (X)$ . We also give a characterisation of two-dimensional spaces with hexagonal norms. Finally, we indicate some new upper estimates concerning $\mu (l_p)$ and $\mu _p(l_p)$ .

Concurrent and parallel chords of spheres in normed linear spaces

2010 ◽  
Vol 47 (4) ◽  
pp. 505-512
Author(s):  
Horst Martini ◽  
Senlin Wu
Keyword(s):  
Banach Spaces ◽  
Unit Sphere ◽  
Normed Linear Space ◽  
Inner Product ◽  
Product Spaces ◽  
Euclidean Case ◽  
Linear Spaces ◽  
Inner Product Spaces ◽  
Finite Dimensional

We give a geometric characterization of inner product spaces among all finite dimensional real Banach spaces via concurrent chords of their spheres. Namely, let x be an arbitrary interior point of a ball of a finite dimensional normed linear space X. If the locus of the midpoints of all chords of that ball passing through x is a homothetical copy of the unit sphere of X, then the space X is Euclidean. Two further characterizations of the Euclidean case are given by considering parallel chords of 2-sections through the midpoints of balls.

scholarly journals Functional Equations Related to Inner Product Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Choonkil Park ◽  
Won-Gil Park ◽  
Abbas Najati
Keyword(s):  
Functional Equation ◽  
Banach Spaces ◽  
Functional Equations ◽  
Vector Spaces ◽  
Inner Product ◽  
Ulam Stability ◽  
Product Spaces ◽  
Real Vector ◽  
Inner Product Spaces

LetV,Wbe real vector spaces. It is shown that an odd mappingf:V→Wsatisfies∑i−12nf(xi−1/2n∑j=12nxj)=∑i=12nf(xi)−2nf(1/2n∑i=12nxi)for allx1,…,x2n∈Vif and only if the odd mappingf:V→Wis Cauchy additive. Furthermore, we prove the generalized Hyers-Ulam stability of the above functional equation in real Banach spaces.

scholarly journals Stability of the Fréchet Equation in Quasi-Banach Spaces

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 490
Author(s):  
Sang Og Kim
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Spaces ◽  
Point Theorem ◽  
Inner Product ◽  
Main Role ◽  
Product Spaces ◽  
Inner Product Spaces ◽  
Fréchet Equation

We investigate the Hyers–Ulam stability of the well-known Fréchet functional equation that comes from a characterization of inner product spaces. We also show its hyperstability on a restricted domain. We work in the framework of quasi-Banach spaces. In the proof, a fixed point theorem due to Dung and Hang, which is an extension of a fixed point theorem in Banach spaces, plays a main role.

scholarly journals On 2-Inner Product Spaces and Reproducing Kernel Property

2019 ◽  
Author(s):  
Saeed Hashemi Sababe
Keyword(s):  
Machine Learning ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Reproducing Kernels ◽  
Inner Product ◽  
Product Spaces ◽  
Inner Product Spaces

This paper is devoted to the study of reproducing kernels on 2-inner product Hilbert spaces. We focus on a new structure to produce reproducing kernel Hilbert and Banach spaces. According to multi-variable computing, this structures can be useful in electrocardiographs, machine learning and economy

scholarly journals Fuzzy Inner Product Space: Literature Review and a New Approach

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 765
Author(s):  
Lorena Popa ◽  
Lavinia Sida
Keyword(s):  
Literature Review ◽  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Product Spaces ◽  
Comprehensive Overview ◽  
New Approach ◽  
Inner Product Spaces ◽  
Product Function ◽  
Suitable Definition

The aim of this paper is to provide a suitable definition for the concept of fuzzy inner product space. In order to achieve this, we firstly focused on various approaches from the already-existent literature. Due to the emergence of various studies on fuzzy inner product spaces, it is necessary to make a comprehensive overview of the published papers on the aforementioned subject in order to facilitate subsequent research. Then we considered another approach to the notion of fuzzy inner product starting from P. Majundar and S.K. Samanta’s definition. In fact, we changed their definition and we proved some new properties of the fuzzy inner product function. We also proved that this fuzzy inner product generates a fuzzy norm of the type Nădăban-Dzitac. Finally, some challenges are given.

Triangle Congruence Characterizations of Inner Product Spaces

1989 ◽  
Vol 144 (1) ◽  
pp. 81-86
Author(s):  
Charles R. Diminnie ◽  
Edward Z. Andalafte ◽  
Raymond W. Freese
Keyword(s):  
Inner Product ◽  
Product Spaces ◽  
Inner Product Spaces
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