Isosceles Orthogonal Triples in Linear 2-Normed Spaces

✍ Scribed by Y. J. Cho; C. R. Diminnie; R. W. Freese; E. Z. Andalafte

• Publisher: John Wiley and Sons
• Year: 1992
• Tongue: English
• Weight: 457 KB
• Volume: 157
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19921570118

✦ Synopsis

Abstract

A triple (x, y, z) in a linear 2‐normed space (X, ‖.,.‖) is called an isosceles orthogonal triple, denoted |(x, y, z), if

|(.,.,.) is said to be homogeneous if |(x, y, z) implies |(ax, y, z) for all real a and it is additive if |(x~1~, y, z) and |(x~2~, y, z) imply that |(x~1~ + x~2~, y, z). In addition to developing some basic properties of |(.,.,.), this paper shows that under the assumption of strict convexity, every subspace of X of dimension ≤ 3 contains an isosceles orthogonal triple. Further, if (X, ‖.,.‖) is strictly convex and |(…,.) is either homogeneous or additive, then (X, ‖.,.‖) is a 2‐inner product space.