On parallelogram areas in normed linear spaces
type: Item
instance of: scholarly article
title: On parallelogram areas in normed linear spaces
zbMATH Open document ID: 1112.39018
author: Claudi Alsina
author: Maria Santos Tomás
published in: Aequationes Mathematicae
publication date: 2007-02-14T00:00:00Z
review text: The author offers characterizations of inner product spaces in terms of equations satisfied by areas of parallellograms defined among others in the following different ways: \[ \begin{aligned} A_1(x,y)&:=(\| x\|^2\| y|^2-r(x,y)r(y,x)\,)^{1/2},\qquad r(x,y):=\lim_{t\to+0}(\| x+ty\|^2-\| x\|^2)/(2t); \\ A_2(x,y)&:=(\|\:(x-y)y+\,(x-y)^2(\| y\|^2-r(x,y)\,)\| x-y\|^{-2}\: \|; \\ A_3(x,y)&:=\| x-y\|(\,\| x\|^2-\| x-y\|(1-(\| y\|^2-r(x,y)\,)\| x-y\|^{-2}\,)\,)^{1/2}.\end{aligned} \] One result offered is that a real normed linear space \((E,\|\;\|)\;(\dim E\geq 2)\;\) is an inner product space iff (at least) one (thus all) of the \(A_j\)'s satisfies \(A_j(z,x+y)^2+A_j(z,x-y)^2=2A_j(z,x)^2+2A_j(z,y)^2\quad(j=1,2,3)\) for all \(x,y,z\) in \(E.\)
reviewed by: János Aczél
Mathematics Subject Classification ID: 39B52
Mathematics Subject Classification ID: 46C15
zbMATH DE Number: 5126567
zbMATH Keywords: normed linear spaces
zbMATH Keywords: inner product spaces
zbMATH Keywords: functional equations
zbMATH Keywords: parallelogram areas
MaRDI profile type: Publication
full work available at URL: s00010-005-2814-6
OpenAlex ID: W2160756337
DOI: 10.1007/S00010-005-2814-6
description: scientific article; zbMATH DE number 5126567