Positivity 8: 143–164, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Subspaces of Normed Riesz Spaces
ONNO VAN GAANS
Department of Applied Mathematical Analysis, Faculty ITS, Delft University of Technology, P.O.
Box 5031, 2600 GA Delft, The Netherlands. (E-mail: email@example.com)
(Received 14 May 2002; accepted 27 August 2002)
Abstract. 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 px py for all x and y with −y x 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.
Subject Classiﬁcations (1991 MSC): 46B40, also 46A40, 46B28
Key words: embedding, M-norm, monotone seminorm, normed Riesz space, norm dual, partially
ordered vector space, space of operators
Not every linear subspace of a Riesz space (vector lattice) is a Riesz space. In fact,
Luxemburg showed in  that every partially ordered vector space can linearly,
bipositively be imbedded in a Riesz space. Recall that a linear map AE → F ,
where E and F are partially ordered vector spaces, is called bipositive if for any
x ∈E one has Ax 0 ⇐⇒ x 0. In the study of normed ordered spaces a
similar problem is to determine what spaces are the subspaces of normed Riesz
spaces (that is, Riesz spaces with Riesz norms). We present a characterization of
THEOREM 1.1. A partially ordered vector space E with a norm p can linearly,
bipositively, isometrically be embedded in a normed Riesz space if and only if the
positive cone E
is closed and pxpy for all xy ∈ E with −y x y
A minor adaptation yields a similar theorem for embedding in M-normed Riesz
spaces. With aid of Theorem 1.1 one can deduce Krein’s lemma on the directedness
of the norm dual of a monotonely normed partially ordered vector space directly
from the much more straightforward result for normed Riesz spaces.
Apart from the embedding, the property of the norm in Theorem 1.1 shows
useful in other respects as well. To exploit its use in the theory of spaces of