Duality for Locally Compact Abelian Groups

  • Dinakar Ramakrishnan
  • Robert J. Valenza
Part of the Graduate Texts in Mathematics book series (GTM, volume 186)


For a locally compact abelian group G, its group Ĝ of characters (i.e., continuous homomorphisms from G to S 1) also acquires the structure of a topological group. In this chapter, we give two distinctive characterizations of what turns out to be the same underlying topology for Ĝ and examine this topology in detail. The main result is the Pontryagin duality theorem, which says in effect that G and Ĝ are mutually dual, both algebraically and topologically. To prove this, we build upon the results of the previous chapter, especially insofar as the introduction of functions of positive type makes a critical correspondence with the theory of unitary representations.


Compact Subset Open Neighborhood Haar Measure Radon Measure Compact Abelian Group 
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Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Dinakar Ramakrishnan
    • 1
  • Robert J. Valenza
    • 2
  1. 1.Mathematics DepartmentCalifornia Institute of TechnologyPasadenaUSA
  2. 2.Department of MathematicsClaremont McKenna CollegeClaremontUSA

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