The Banach--Mazur Theorem for Spaces with Asymmetric Norm

P. A. Borodin

Matematičeskie zametki, Tome 69 (2001) no. 3, pp. 329-337

Résumé

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 $F\colon K\to\mathbb R$.