ABSTRACT
LetT E 'l'(fl). Then there is a bounded neighborhood ofO in 'l(O) which is mapped by T into the unit disc in C. By Theorem 1.8, that means there is an integer m E N0 , a compact set K C 0 and a positive number r such that the neighborhood of 0 in '&(0) defined by
U ~ {<J> E ~(0) 'Pm.K(<J>) < '} satisfies \T(<fl)l ,; I for every 4> E U. If<!> E 'l(O) and Pm . .ct$) = 0, then A$ E U for every A > 0 and \T(A<P)I = AfT($)\ ,;;; I. Therefore !T<If>)l ..;:,;: 1/A for every A > 0, which means that T(lf>) = 0. Since Pm.rllf>) = 0 for every 4> E \1!(0 - K) we conclude that T = 0 on 0 - K, that is, supp T C K. 0
Example 3.1 The sequence Tn = ~7 d'&t. with a> 0. converges in Qll'(R) but not in ~'(R). To see that, let 4> E '2ll(R). Then there exists an integer m such that 4> = 0 outside [- m,m], and
we choose the test function lf>(x) = a-x E ~(R) then (Tn,lf>) = ~~ a*a-k = n"'"""' oo. Thus the infinite sum Lf ak&k lies in £ll'(R) but not in '\f;'(R).
