The Banach-Mazur theorem for spaces with asymmetric norm and its applications in convex analysis

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 \ This assertion is used to obtain nontrivial representations of an arbitrary convex closed body M subset of R-n, an arbitrary compact set K subset of R-n, and an arbitrary continuous function F: K --> R.