Subspaces of normed Riesz spaces

Summary: It will be shown that a normed partially ordered vector space is linearly, norm, and order isomorphic to a subspace of a normed Riesz space if and only if its positive cone is closed and its norm $p$ satisfies $p(x)\le p(y)$ for all $x$ and $y$ with $-y \le x \le y$. A similar characterization of the subspaces of $M$-normed Riesz spaces is given. With aid of the first characterization, Krein's lemma on directedness of norm dual spaces can be directly derived from the result for normed Riesz spaces. Further properties of the norms ensuing from the characterization theorem are investigated. Also a generalization of the notion of Riesz norm is studied as an analogue of the $r$-norm from the theory of spaces of operators. Both classes of norms are used to extend results on spaces of operators between normed Riesz spaces to a setting with partially ordered vector spaces. Finally, a partial characterization of the subspaces of Riesz spaces with Riesz seminorms is given.