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Mat. Zametki, 2001 Volume 69, Issue 3, Pages 329–337 (Mi mzm506)

This article is cited in 24 papers

The Banach–Mazur Theorem for Spaces with Asymmetric Norm

P. A. Borodin

M. V. Lomonosov Moscow State University

Abstract: We establish an analog of the Banach–Mazur theorem for real separable linear spaces with asymmetric norm: every such space can be linearly and isometrically embedded in the space of continuous functions $f$ on the interval $[0,1]$ equipped with the asymmetric norm $\|f|=\max\{f(t)\colon t\in[0,1]\}$. This assertion is used to obtain nontrivial representations of an arbitrary convex closed body $M\subset\mathbb R^n$ , an arbitrary compact set $K\subset\mathbb R^n$, and an arbitrary continuous function $F\colon K\to\mathbb R$.

UDC: 517.982

Received: 24.01.2000

DOI: 10.4213/mzm506


 English version:
Mathematical Notes, 2001, 69:3, 298–305

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