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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

The transitive property of parallel lines is a characteristic property of real strictly convex Banach spaces


Author: J. E. Valentine
Journal: Proc. Amer. Math. Soc. 95 (1985), 604-606
MSC: Primary 51K05; Secondary 46B20
MathSciNet review: 810171
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Abstract: In a recent paper Freese and Murphy said a complete, convex, externally convex metric space has the vertical angle property provided for each four of its distinct points $ p$, $ q$, $ r$, $ s$, if $ m$ is a midpoint of $ p$ and $ q$ and of $ r$ and $ s$, then $ pr = qs$. In this paper we say a line $ L$ is parallel to a line $ N$ in such a space provided $ L$ and $ N$ contain points $ p$, $ r$, and $ q$, $ s$, respectively, such that the segments $ S\left( {p,q} \right)$ and $ S\left( {r,s} \right)$ have a common midpoint $ m$. We further assume that if line $ L$ is parallel to line $ N$ and line $ N$ is parallel to line $ R$, then $ L$ is parallel to $ R$. The main result of this paper is that such a space is a real strictly convex Banach space. Since real strictly convex Banach spaces have all of the above properties, the characterization is then complete.


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DOI: https://doi.org/10.1090/S0002-9939-1985-0810171-2
Keywords: Banach space, convex, externally convex, metric space, strictly convex, transitive property of parallel lines, vertical angle property
Article copyright: © Copyright 1985 American Mathematical Society