proceedings of the american mathematical society Volume 110, Number 1, September 1990 ON THE TOPOLOGY OF THE SPACE OF CONVOLUTION OPERATORS IN K'M SALEH ABDULLAH (Communicated by Palle E. T. Jorgensen) Abstract. In this paper we show that on the space 0'c(K'M : K'M) of convolu- tion operators on KM , the topology rb of uniform convergence on bounded subsets of KM is equal to the strong dual topology. Introduction In previous work (see [1, 2]) we redefined the space K'M of rapidly increasing distributions and the space 0'c(K'M: K'M) of its convolution operators. The space 0'C(K'M: K'm) was provided with several topologies, the topology xh of uniform convergence on bounded subsets of KM , the topology of x'b of uniform convergence on bounded subsets K'M , the projective limit topology x where 0'c(K'M: K'M) was considered as the projective limit of the spaces e~ X'S', and the strong dual topology where 0'c(K'M: K'm) was considered as the dual of Oc(KM: KM) (see definitions below). It was shown in [2] that xb and x'b are equal, and x is equal to the strong dual topology. The question whether xb and the strong dual topology are equal or not was left unanswered in [2]. Our main result in this paper is the following. Theorem. On the space 0'C(K'M:K'm) the topology xh is equal to the strong dual topology. To establish this result we need the following Lemma. The convolution map from (0'c,xh)xOc into Oc is separately contin- uous. To avoid lengthy proofs we will present these results in several steps. Received by the editors October 17, 1989; this paper has been presented to the Society at the Phoenix meeting on January 12, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 46F05, 46F10. The presentation was sponsored by the University of Kuwait. ©1990 American Mathematical Society 0002-9939/90 $1.00+ $.25 per page 177 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use