1. ABSTRAC T HARMONIC ANALYSIS: A SURVEY 7

and for w £ 5, the w-th component of as is the same as that of a. This allows us

to write

x{o)=nx^'x^5)=n

^(^)=n^^

ves ves v

because Xv{&v) — 1 for all v (£ S. This proves the lemma.

As a complementary statement to the above lemma we also have

LEMM A 2. Suppose that \v £ Homc(GVj

Gx)

is given for each v E S with Xv

trivial on Hv for almost all v. Then the product

V

defines a quasi-character in Homc(G,Cx).

Proof, x

is

clearly a homomorphism. To prove continuity we select a set S contain-

ing all v where Xv(Hv) ^ 1. Let s be the cardinality of 5. Given a neighborhood U

of 1 in C x we can select another neighborhood V of 1, possibly smaller, such that

Vs C U, i.e. any product of 5 elements of V is an element in U. For each v choose

a neighborhood Nv of 1 in Gv such that Xv(Nv) C V for all v G S and Nv — Hv

for all v ^ S. Then clearly

X([[NV)CVSCU.

V

This proves the lemma.

We observe that a quasi-character x = 11^ Xv of G is a character if and only

if all the local components Xv are also characters. Now let H^ be the subgroup of

the dual group Gv consisting of the characters of Gv which are trivial on Hv. Since

Hv is compact by Pontrjagin duality, its dual group Hv = G/H^ is discrete and

hence H^ is open; similarly, since Hv is open and compact, Gv/Hv is discrete and

its dual group (GV/HV)A = H^ is compact.

We are now able to view the dual group of G as the restricted direct product

of the dual groups Gv with respect to the subgroups H^. In fact the product

decomposition \ — EL Xv gives an algebraic isomorphism. We state this formally.

T H E O R E M 7. The factorization x — Ylv Xv of a quasi-character gives an

algebraic and topological isomorphism

Hvmc([(Gv,T) ^ Jj'ffomc(Gt,,T),

V V

where the first product Y[v Gv denotes the restricted direct product of the {Gv} with

respect to the subgroups {Hv} and the second is the restricted direct product of the

groups {Homc(Gv,T)} with respect to the subgroups {H^}.

Proof. We need only verify that indeed we have a topological isomorphism. To

say that x — Ylv Xv is close to 1 means that x is close to 1 for a large compact

subset B = Yiv By of G where Bv C Gv is compact and Bv = Hv for almost all