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 Mat. Zametki, 1976, Volume 19, Issue 2, Pages 237–246 (Mi mz7743)

Defining a metric in a linear space by means of a family of subsets

A. I. Vasil'ev

Institute of Mathematics and Mechanics, Ural Scientific Center of the AS of USSR

Abstract: Necessary and sufficient conditions are given on a family $\{A_r\}_{r>0}$ of subsets of a real linear space $X$ under which $\inf\{r>0:x\in A_r\}$ is a quasinorm [1] on X. It is shown that for any symmetric (about zero) closed set $A$ in a normed space $X$ containing the ball $\{x\in X:\|x\|\le1\}$ there exists a quasinorm $|\cdot|$ on $X$ such that $A=\{x\in X:\|x\|\le1\}$. Examples are constructed of linear metric spaces in which there exists a Chebyshev line which is not an approximately compact set.

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English version:
Mathematical Notes, 1976, 19:2, 141–145

Bibliographic databases:

UDC: 513.8

Citation: A. I. Vasil'ev, “Defining a metric in a linear space by means of a family of subsets”, Mat. Zametki, 19:2 (1976), 237–246; Math. Notes, 19:2 (1976), 141–145

Citation in format AMSBIB
\Bibitem{Vas76} \by A.~I.~Vasil'ev \paper Defining a~metric in a~linear space by means of a~family of subsets \jour Mat. Zametki \yr 1976 \vol 19 \issue 2 \pages 237--246 \mathnet{http://mi.mathnet.ru/mz7743} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=410317} \zmath{https://zbmath.org/?q=an:0334.46003} \transl \jour Math. Notes \yr 1976 \vol 19 \issue 2 \pages 141--145 \crossref{https://doi.org/10.1007/BF01098747}