proceedings of the american mathematical society Volume 111, Number 4. April 1991 ON THE CONVERGENCE IN A' STEVANPILIPOVIC (Communicated by Palle E. T. Jorgensen) Abstract. We prove the following assertion: Let T,, j € N , be a sequence in A such that 7" * <fr converges to 0 in A as ;'-»«, for any 4> 6 3! . Then T' • —► 0 in S? as j -* oo. The following problem was presented by A. Kamiñski: Let Tj be a sequence in jA such that F * r/> converges to 0 in ¿A as y —> oo for any <fr e <A. Does F —► 0 in A' as j-»oo? K. Keller gave a positive answer to this question in [3] by using an original method based on a theorem of Grothendieck. But the assertion follows directly from the equality of the linear hull of <A * AA, \in(SAAAAA?), and of S" proved by J. Voigt [9] and even earlier by H. Petzeltová and P. Vrbová [4]. Note that lin(^" * 2A) = 3A is known from the following papers: L. A. Rubel, W. A. Squires, and B. A. Taylor [5], and J. Dixmier and P. Malliavin [1]. The aim of this paper is to prove the following theorem: Theorem 1. Let F , j e N, be a sequence from jA such that F * <pconverges to 0 in 5A' as j —► oo for any 4> eA3. Then T',—► 0, j —> oo, in jA . The well-known result of Schwartz should be mentioned: "If T e 3' and r^eA for any tj> € 9S, then reA ." In the proof we shall use Keller's method [3], and since it is not known whether lin(^ *¿A is equal to A7> t this method is essential in our formulation of the problem. As usual, N is the set of strictly positive integers, N0 = N U {0} , and Z = N0 u (-N). We denote by 2¡, S?, and 3¡K , where F is a compact subset of the Euclidean space R" (K C R"), the well-known Schwartz testing-function spaces. As in [8, Chapter I, §5], ¿Apk, k e N0, is the completion of A7> in the norm Received by the editors February 9, 1989 and, in revised form, April 30, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 46F05. Key words and phrases. Tempered distributions. ©1991 American Mathematical Society 0002-9939/91 $1.00+ $.25 per page 949 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use