  The aim of this paper is to study the basic properties of the Thompson metric
$d_T$ in the general case of a real linear space $X$ ordered by a cone $K$. We
show that $d_T$ has monotonicity properties which make it compatible with the
linear structure. We also prove several convexity properties of $d_T$ and some
results concerning the topology of $d_T$, including a brief study of the
$d_T$-convergence of monotone sequences. It is shown most of the results are
true without any assumption of an Archimedean-type property for $K$. One
considers various completeness properties and one studies the relations between
them. Since $d_T$ is defined in the context of a generic ordered linear space,
with no need of an underlying topological structure, one expects to express its
completeness in terms of properties of the ordering, with respect to the linear
structure. This is done in this paper and, to the best of our knowledge, this
has not been done yet. The Thompson metric $d_T$ and order-unit (semi)norms
$|\cdot|_u$ are strongly related and share important properties, as both are
defined in terms of the ordered linear structure. Although $d_T$ and
$|\cdot|_u$ are only topological (and not metrical) equivalent on $K_u$, we
prove that the completeness is a common feature. One proves the completeness of
the Thompson metric on a sequentially complete normal cone in a locally convex
space. At the end of the paper, it is shown that, in the case of a Banach
space, the normality of the cone is also necessary for the completeness of the
Thompson metric.

Functional Analysis