Characters and duality of locally compact Abelian groups

Abstract

This chapter is devoted to what might be called the “fine structure” of locally compact Abelian groups. We will obtain extremely detailed information about the structure of these groups. The basic tool in this program is the character group, which is defined and studied in §23. The character group is important in large measure because of the Pontryagin-van Kampen duality theorem, which is stated, proved, and utilized in §24. In §25, we apply the duality theorem to study a number of special locally compact Abelian groups, and in §26 we apply it to certain problems in structure theory and analysis.