1. ABSTRAC T HARMONIC ANALYSIS: A SURVEY 5

THEOREM 6. (Plancherel) The Fourier transform &*-& is an isometry of

Ll(G)

fl

L2{G)

onto a dense subspace of

L2(G);

hence it has an extension to an

isometry of

L2(G)

onto

L2(G).

The point of view which we adopt later on is that the appropriate setting for

discussing the Fourier transform is the theory of distributions. From this perspec-

tive, the most important property of the Fourier transform is a generalization of the

classical fact that homogeneous distributions (in the sense of Gelfand) are mapped

to homogeneous distributions.

Restricted Direct Products

Let 5 denote an index set with an infinite number of elements. For each v G S

we let Gv be a locally compact abelian group and for almost all i, meaning for all

but a finite number of v, Hv is a given subgroup of Gv, which is open and compact.

The restricted direct product of the {Gv} with respect to the subgroups {Hv},

consists set theoretically of all those elements g = (gv) in the cartesian product

Y[v Gv, made up of "vectors" whose components gv in Gv are in Hv for almost all

v. This group will be denoted in the following by G. To introduce a topology on

G, we consider for each subset S of S, which includes those exceptional v where

the Hv are not defined, the group

Gs=l[Gvx ]JHV.

veS v(£S

This group is also denoted by G(5); Gs with its associated Tychonoff topology is

a locally compact abelian group. It is clear that any element of G belongs to such

a subgroup Gs for a suitable finite set S; hence

G = U

5

G

5

,

where the union is taken over all finite subsets S of S which contain those v where

the Hv are not defined. To assign a topology to G, we prescribe as a fundamental

system of neighborhoods of 1 in G, the set of neighborhoods of 1 in Gs-

Since all the subgroups Hv are open by hypothesis, it is clear from the definition

of the Tychonoff topology on the group G5, that the family of all open subsets of

the form

N =

J\NV,

V

where Nv is a neighborhood of 1 in Gv and Nv = Hv for almost all v form a

fundamental system of neighborhoods of 1 in G. In essence the above construction

reduces the study of the topological properties of the restricted direct product G

to that of the subgroups Gs •

In this chapter we will identify the group Gv with a subgroup of G via the

embedding

iv

:

Gv

— G

given by iv(gv) = (gw)-, where gw = gv if w = v and gw = 1 otherwise.

In several arguments concerning G, it will be convenient to think of the sub-

group Gs as a finite product of locally compact abelian groups and of a compact