One of the principal methods used in mathematics to represent complex objects involves the application of certain operations to finite or especially infinite sets of simpler objects which can be used to essentially approximate the complex objects. Classical examples of such methods include， among many others， infinite series representations of functions in mathematical analysis and series expansions with respect to a Schauder basis in the study of separable Banach spaces in functional analysis.
　　The author of this monograph is a prominent expert in the study of Schauder bases， but the present book is devoted to a different kind of application of the approach described above， namely to the representation of complex objects through the operation of taking suprema （or infima） of families of elements in an ordered space.
　　In the late 1960s and early 1970s it was realized that it is possible， and relatively straightforward， to obtain many of the principal results of convex duality theory using the representation of a convex function as the point wise supremum of the set of its affine minorants. Moreover， many of these results do not depend on the linear structure of the class of minorizing functions， The corresponding observation for closed convex sets was made even earlier， that is， many results for these sets easily follow from their \"outer\" representation as intersections of closed half-spaces. Also， many of these results can be generalized to sets which can be represented as the intersections of other families of sets that are not necessarily half-spaces.