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,*).
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