1. Abstract Harmonic Analysis: A Survey
Introduction. In this section, which is purely of a utilitarian nature, we
gather all the relevant definitions and results from abstract harmonic analysis that
are needed for the study of zeta functions on number fields. After a brief review of
some basic topological notions regarding locally compact abelian groups we recall
the notions of Haar measure, Pontrjagin duality, and Fourier transforms. The
central topic of these introductory notes is a discussion of the Poisson summation
formula. The reader who is familiar with these notions may want to proceed to
section 1.2 where the study of zeta and L-functions begins in earnest.
Basic Facts
Let {Xv}v€s be a family of topological spaces indexed by the set S. For each
v G S let Tv be the collection of open sets in Xv. The cartesian product of {Xv}ves
is denoted by
and as usual, it is identified with the set of all functions
such that x(v) G Xv\ for simplicity, the elements of ELes ^v a r e denoted by x
(xv)ves- A projection mapping is defined as follows:
prv : Y\XV - Xv, x = (xv), prv(x) = xv.
A topology on \\XV is desired with the property that all projection mappings
be continuous functions. Such a topology is provided by the Tychonoff topology
whereby a basis for the open sets is given by the family of subsets of the form
Y[v Uv, where Uv is an open subset in Tv and Uv Xv for almost all v, i.e. all
v G S except a finite number; this is the weakest topology for which the projection
mappings prv are continuous. The basic result about this topology is the following:
THEOREM 1. (Tychonoff) The product space
YlvXv, ™ith
respect to the Ty-
chonoff topology, is compact if and only if each of the spaces Xv is compact.
DEFINITION 1. A topological space X is called locally compact if for each point
x G X there is an open set Ux such that x G Ux and the closure Ux is compact.
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