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

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.