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 Mat. Zametki, 2018, Volume 104, Issue 1, Pages 118–130 (Mi mz11668)

A Sublinear Analog of the Banach–Mazur Theorem in Separable Convex Cones with Norm

F. S. Stonyakin

Crimea Federal University, Simferopol

Abstract: A special class of separable normed cones, which includes convex cones in normed spaces and in spaces with an asymmetric norm, is distinguished on the basis of the functional separability of elements. It is shown that, generally, separable normed cones admit no linear injective isometric embedding in any normed space. An analog of the Banach–Mazur theorem on a sublinear injective embedding of a separable normed cone in the cone of real nonnegative continuous functions on the interval $[0;1]$ with the ordinary sup-norm is obtained. This result is used to prove the existence of a countable total set of bounded linear functionals for a special class of separable normed cones.

Keywords: separable normed cone, space with asymmetric norm, Hahn–Banach theorem, Banach–Mazur theorem, sublinear injective isometric embedding, total set of bounded linear functionals.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation ÌÊ-176.2017.1 The work was financially supported by a grant of the President of the Russian Federation for State Support of Young Russian Scientists – Candidates of Science (project code MK-176.2017.1).

DOI: https://doi.org/10.4213/mzm11668

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English version:
Mathematical Notes, 2018, 104:1, 111–120

Bibliographic databases:

UDC: 517.98
Revised: 14.07.2017

Citation: F. S. Stonyakin, “A Sublinear Analog of the Banach–Mazur Theorem in Separable Convex Cones with Norm”, Mat. Zametki, 104:1 (2018), 118–130; Math. Notes, 104:1 (2018), 111–120

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz11668
• https://doi.org/10.4213/mzm11668
• http://mi.mathnet.ru/eng/mz/v104/i1/p118

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. F. S. Stonyakin, “Hahn–Banach type theorems on functional separation for convex ordered normed cones”, Eurasian Math. J., 10:1 (2019), 59–79
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