Convex Sets and Sublinearity

Abstract

This chapter deals with the basic theory of sublinear functions. We begin with some preliminary remarks about ordering and give a short survey about the basic facts on sublinear functions. Then the proof of the algebraic version of the Hahn-Banach theorem follows and the section ends with the separation theorem for convex sets in vector spaces. In the next chapter topological vector spaces are considered, and the Hahn-Banach theorem together with the separation theorem are stated for locally convex vector spaces. Throughout this book we will only consider real vector spaces and will use the notions of linear function and linear functional simultaneously. We follow the representations given in the textbook of F. Hirzebruch and W. Scharlau [51] and in the textbook of S. Rolewicz [86].