250 XVII DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
have by definition, for any u e ^(RK)>

<S, uy = f dT(x) tf(y)u(x + y) dk(y) = <T ® A, g\

J J

where A is Lebesgue measure on Rn, and g(x, y) =f(y)u(x + j>). However,

(f(y}u(x

hence <S? w> = <T ® (w • A), /z>, where h(x9 y) =f(y - x); and therefore
<S , w> = uGO JAOO (f(y - x) dT(x\

(

which proves (1 7.1 2.1 .1 ).
In particular :

(17.12.2) If T e <T(Rn) owd /e <^(Rn), or z/ T e ^'(

T */e <f(Rn). If T 6 <T(Rn) a«J z/ (/p) z,s a sequence of functions in <?(RM)

converging to 0 in ^/z/^ ^pace, ^/ze« the sequence (T */p) rents' to 0 zn

Let A be the (compact) support of T, let V be a compact neighborhood of
A, and let K be any compact subset of RM. If we put gp(x, y) =fp(x - y), then
the sequence of elements gp(x, •) of <f(Rn) converges to 0 in this space,
uniformly with respect to x e K. For the partial derivative with multi-index v
of the function y *-»gp(x, y) is (- l)lv|Dv/p(;c - j;). Since the sequence (Dv/p(z))
converges uniformly to 0 in the compact set K + (— V), the sequence of
functions (x, y) H» Dv/p(x — y) converges uniformly to 0 on K x V; hence, by
the definition of distributions, the sequence of functions x\-^hp(x) =

fp(x — y) dT(y) converges uniformly to 0 in K, and the same is true of the

sequence of partial derivatives x*-+Dvhp(x) for any multi-index v, by virtue of
(17.10.1).

\

(17.12.3) For each open subset U o/R", the set ®(U) of C°° -functions on U
with compact support, identified with a space of distributions on U (17.5.3),
is weakly dense in @'(U).

Let (Km) be a fundamental sequence of compact subsets of U, and let
hm be a function belonging to ®(U) which is equal to 1 on KM . Since for each
u e ^(U) the reexists an integer m such that Km is a neighborhood of the
support of u, it follows that for each distribution T e &>'(U) we havea function in $(p~"m\Rn) such that, for each