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.

UDC: 513.8

Received: 17.07.1974

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

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