1. ABSTRACT HARMONIC ANALYSIS: A SURVEY

9

(i) H is generated by a compact neighborhood of the identity of G (hence it is

closed and open in G).

(ii) H' is a compact subgroup of H and Hj'Hf is isomorphic to an elementary

subgroup.

To each pair (i7, H') we associate the family S(H, H') of continuous functions

on G supported on H, constant on cosets of H modulo

Hf

and such that the

function on H/H' which results by restriction to H and passage to the quotient

gives a function on

S(H/Hf).

The space S(G) is then defined as the union of the

spaces S(H, H') and its topology is that induced by the inductive limit of those of

S(H/H'). In this topology a convex subset X is a neighborhood of the identity in

S{G) if, for any pair (H, if'), the image XnS(H, H') in S(H/H') is a neighborhood

of 1 in S{H/H'). The space S(G) is a complete, locally convex topological space

with a countable base of neighborhoods. In S(G) there is a correspondence between

the bounded sets and the relatively compact sets.

DEFINITIO N 7. The space

Sf(G)

of tempered distributions on G consists of

the continuous complex valued linear functionals defined on S(G): if {&j}jew is a

sequence of functions in S(G) which converge to 0 in S(G), and ifTe

Sf(G),

then

the sequence of complex numbers

{ T ( $ J ) }

J G

N

also converges to 0.

Below we introduce other spaces of distributions that are more appropriate to

a discussion of vector spaces X over R or over C. We begin by recalling that the

support K of a function $ defined on X is the smallest closed set outside of which

$ vanishes; the support of $ will be denoted by supp$.

DEFINITIO N 8. C™(X) is the set of all C°° -functions with compact support

DEFINITIO N 9. Let { $

n

}

n e N

be a sequence of functions in C£°(X). A topology

is introduced in C£°(X) by prescribing that a sequence ln — 0 if

(i) there is a set H such that supp 3n C H holds for all n,

(ii)

Dp&n(t)

— 0 uniformly in H for all p 0, i.e. for each fixed differential

operator

Dp

of order p, we have

sup

\Dpfn(t)\

— 0 as n-^oo.

tEH

Endowed with this topology C%°(X) is complete in the sense that if £n G C%°(X)

and $

n

- $, then $ G C£°(X).

DEFINITIO N 10. A distribution T is a continuous linear functional on C%°(X).

The space of distributions is denoted by

Tf(X).

The above definition means there is a bilinear pairing

( , ) : P ' ( I ) x C W - C

that is continuous in the second argument, i.e. if T G V{X) and {£n}neN is a

convergent sequence of functions in

CQ°(X)

with limit £, then limn_^00(T, $

n

) =

(T,*).