Ordered vector spaces made their debut at the beginning of the twentieth
century. They were developed in parallel (but from a different perspec-
tive) with functional analysis and operator theory. Before the 1950s ordered
vector spaces appeared in the literature in a fragmented way. Their sys-
tematic study began in various schools around the world after the 1950s.
We mention the Russian school (headed by L. V. Kantorovich and the
Krein brothers), the Japanese school (headed by H. Nakano), the Ger-
man school (headed by H. H. Schaefer), and the Dutch school (headed
by A. C. Zaanen). At the same time several monographs dealing exclu-
sively with ordered vector spaces appeared in the literature; see for in-
stance [55, 56, 71, 75, 89, 91]. The special class of ordered vector spaces
known as Riesz spaces or vector lattices has been studied more extensively;
see the monographs [14, 15, 66, 68, 86, 88, 93].
The theory of ordered vector spaces plays a prominent role in functional
analysis. It also contributes to a wide variety of applications and is an
indispensable tool for studying a variety of problems in engineering and
economics; see for instance [29, 31, 35, 36, 38, 42, 47, 49, 54, 64, 65, 76].
The introduction of Riesz spaces and more broadly ordered vector spaces
to economic theory has proved tremendously successful and has allowed
researchers to answer difficult questions in general price equilibrium theory,
economies with differential information, the theory of perfect competition,
and incomplete assets economies.
The goal of this monograph is to present the theory of ordered vector
spaces from a contemporary perspective that has been influenced by the
study of ordered vector spaces in economics as well as other recent appli-
cations. We try to imbue the narrative with geometric intuition, which is