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 $\mu _p(l_p)$ .

scholarly journals A characterisation of Hilbert spaces via orthogonality and proximinality

2005 ◽  
Vol 71 (1) ◽  
pp. 107-111
Author(s):  
Fathi B. Saidi
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Real Banach Space ◽  
Nonzero Element ◽  
Dimensional Subspace ◽  
Two Dimensional ◽  
Orthogonal Direction ◽  
Open Problems

In this paper we adopt the notion of orthogonality in Banach spaces introduced by the author in [6]. There, the author showed that in any two-dimensional subspace F of E, every nonzero element admits at most one orthogonal direction. The problem of existence of such orthogonal direction was not addressed before. Our main purpose in this paper is the investigation of this problem in the case where E is a real Banach space. As a result we obtain a characterisation of Hilbert spaces stating that, if in every two-dimensional subspace F of E every nonzero element admits an orthogonal direction, then E is isometric to a Hilbert space. We conclude by presenting some open problems.

scholarly journals On the self-length of two-dimensional Banach spaces

1996 ◽  
Vol 53 (1) ◽  
pp. 101-107 ◽  
Author(s):  
B. Chalmers ◽  
C. Franchetti ◽  
M. Giaquinta
Keyword(s):  
Banach Space ◽  
Euclidean Space ◽  
Banach Spaces ◽  
Real Banach Space ◽  
The Self ◽  
Two Dimensional ◽  
Minimum Value

The aim of this paper is to prove the following result: if X is a 2-dimensional symmetric real Banach space, then its self-length is greater than or equal to 2π. Moreover, the minimum value 2π is uniquely attained (up to isometries) by euclidean space.

scholarly journals An algebraic characterization of complete inner product spaces

1986 ◽  
Vol 9 (1) ◽  
pp. 47-53
Author(s):  
Vasile I. Istratescu
Keyword(s):  
Banach Space ◽  
Linear Operators ◽  
Inner Product ◽  
Algebraic Characterization ◽  
Product Spaces ◽  
Bounded Linear Operators ◽  
Multiplicative Property ◽  
Bounded Linear ◽  
Inner Product Spaces

We present a characterization of complete inner product spaces using en involution on the set of all bounded linear operators on a Banach space. As a metric conditions we impose a “multiplicative” property of the norm for hermitain operators. In the second part we present a simpler proof (we believe) of the Kakutani and Mackney theorem on the characterizations of complete inner product spaces. Our proof was suggested by an ingenious proof of a similar result obtained by N. Prijatelj.

scholarly journals The hyperstability of AQ-Jensen functional equation on 2-divisible abelian group and inner product spaces

2015 ◽  
Vol 53 (2) ◽  
pp. 59-72
Author(s):  
Iz-iddine EL-Fassi ◽  
Samir Kabbaj
Keyword(s):  
Banach Space ◽  
Functional Equation ◽  
Abelian Group ◽  
Inner Product ◽  
Product Spaces ◽  
Inner Product Spaces ◽  
Jensen Functional Equation ◽  
Jensen Functional ◽  
Class Of Functions ◽  
Divisible Abelian Group

Abstract In this paper, we prove the hyperstability of the following mixed additive-quadratic-Jensen functional equation $$2f({{x + y} \over 2}) + f({{x - y} \over 2}) + f({{y - x} \over 2}) = f(x) + f(y)$$ in the class of functions from an 2-divisible abelian group G into a Banach space.

A Generalization of Uniformly Rotund Banach Spaces

1979 ◽  
Vol 31 (3) ◽  
pp. 628-636 ◽  
Author(s):  
Francis Sullivan
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Boolean Algebra ◽  
Banach Spaces ◽  
Real Banach Space ◽  
Two Dimensional ◽  
Geometrical Characterization ◽  
Arbitrary Space ◽  
Image Position

Let X be a real Banach space. According to von Neumann's famous geometrical characterization X is a Hilbert space if and only if for all x, y ∈ XThus Hilbert space is distinguished among all real Banach spaces by a certain uniform behavior of the set of all two dimensional subspaces. A related characterization of real Lp spaces can be given in terms of uniform behavior of all two dimensional subspaces and a Boolean algebra of norm-1 projections [16]. For an arbitrary space X, one way of measuring the “uniformity” of the set of two dimensional subspaces is in terms of the real valued modulus of rotundity, i.e. for The space is said to be uniformly rotund if for each 0 we have .

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 Stability and hyperstability of a quadratic functional equation and a characterization of inner product spaces

2018 ◽  
Vol 51 (1) ◽  
pp. 295-303
Author(s):  
Gwang Hui Kim ◽  
Iz-iddine El-Fassi ◽  
Choonkil Park
Keyword(s):  
Banach Space ◽  
Functional Equation ◽  
Quadratic Functional Equation ◽  
Inner Product ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Product Spaces ◽  
Inner Product Spaces ◽  
Class Of Functions

AbstractWe have proved theHyers-Ulam stability and the hyperstability of the quadratic functional equation f (x + y + z) + f (x + y − z) + f (x − y + z) + f (−x + y + z) = 4[f (x) + f (y) + f (z)] in the class of functions from an abelian group G into a Banach space.

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