Results with the Hahn-Banach Theorem

Abstract

This chapter is devoted to several results the proofs of which are based on a theorem known as the Hahn-Banach theorem (due to H. Hahn, 1927, and S. Banach, 1929). Most of these results, as well as the Hahn-Banach theorem itself, are extension theorems dealing with extension of a (linear) operator from a linear subspace to the entire space, thereby preserving certain properties of the operator (such as, for example, positivity or norm boundedness). To prove the Hahn-Banach theorem it is necessary to accept a certain axiom about partially ordered sets, called Zorn’s lemma. We formulate the axiom. The definition of a chain and of a maximal element in a partially ordered set are given in section 1.