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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The transitive property of parallel lines is a characteristic property of real strictly convex Banach spaces
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by J. E. Valentine PDF
Proc. Amer. Math. Soc. 95 (1985), 604-606 Request permission

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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Additional Information
  • © Copyright 1985 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 95 (1985), 604-606
  • MSC: Primary 51K05; Secondary 46B20
  • DOI: https://doi.org/10.1090/S0002-9939-1985-0810171-2
  • MathSciNet review: 810171