Abstract
Necessary and sufficient conditions are given on a familyA r r>0 of subsets of a real linear space X under which infr > 0 ∶ x ∃ A r is a quasinorm [l] on X. It is shown that for any symmetric (about zero) closed set A in a normed space X containing the ball {x ∃ X: ∥x∥ ≤l there exists a quasinorm ¦·¦ on X such that A = {x ∃ X ¦x∶¦ ≤1}. 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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Translated from Matematicheskie Zametki, Vol. 19, No. 2, pp. 237–246, February, 1976.
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Vasil'ev, A.I. Defining a metric in a linear space by means of a family of subsets. Mathematical Notes of the Academy of Sciences of the USSR 19, 141–145 (1976). https://doi.org/10.1007/BF01098747
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DOI: https://doi.org/10.1007/BF01098747