Operator Inequalities (Enhanced Edition‪)‬

In this first chapter functional analytical terms and results are presented
which are needed in the succeeding chapters. Section 1 is concerned with
simple order theoretical concepts. These will be used, for instance, in the
theory of inverse-positive linear operators and its applications in Chapter II.
In Section 2 the order theory of Section 1 is generalized by considering
arbitrary convex sets instead of (order) cones. Since here the approach is
different, Section 2 also provides a new understanding of the order theory. The
results of Section 2 will be used mainly in the theory of linear range-domain
implications in Section IVA. In Section 3 we collect, without proof, a series of
results in functional analysis which are used in existence theories. For
instance, we formulate Schauder's fixed-point theorem, which will be applied
in Sections IlIA, 111.6, V.3, and V.5.