Math. Proc. Camb. Phil. Soc. (1984), 96, 495 495 Printed in Great Britain On A-Mackey convergence in locally convex spaces BY JAN H. FOURIE Department of Mathematics, Potchefstroom University, Potchefstroom 2520, South Africa (Received 12 March 1984) Abstract In this note we introduce the concepts of A-Mackey sequence, A-Mackey con- vergence property, A-Schwartz family and associated A-Schwartz family and con- sider some applications of these to locally convex spaces. Hereby A denotes a Banach sequence space with the AK-property - the results of this paper generalize those in [4] where the case A = I 1 is considered. We obtain a dual characterization of those locally convex spaces which satisfy the A-Mackey convergence property and characterize the dual A-Schwartz spaces in terms of the SM-property which is introduced in [10]. Finally, necessary and sufficient condition for a locally convex space to be ultra- bornological is proved. 1. Introduction Let E be a locally convex space, which we shall throughout tacitly assume to be Hausdorffs. A disc in E is an absolutely convex set in E; if B is a disc in E, we denote by E B the linear subspace \J n nBofE, which is normed by the gauge of B. IfE s happens to be complete, we call B a Banach disc. E is said to be locally complete if its system of bounded subsets has a fundamental system consisting of Banach discs. If A and B are discs in E such that A c B, we denote the continuous inclusion of E A into E B by ^2; ^ has a unique continuous linear extension ^ : £ A -> E B to the completions of the normed spaces. For results and notions in locally convex spaces we refer to [5] and for notions on bornologies on vector spaces to [3]. A sequence of numbers will be denoted by X or (A { ) and (A^) (^ n) will be the sequence |x with fii = 0 for i < n and fi i = A { for i ^ n. Throughout this paper A will denote a Banach sequence space (of scalars) which will at least be a normal symmetric BK-space with AK, containing the sequences with only a finite number of non-zero entries. This means that XeA and \/i t \ < |A^| implies (J.eA, (A ffW )eA and ||(A^)|j = IKA^jJH for all X e A and permutations n of the natural numbers, the co-ordinate projections X ->• X i are continuous and X = 2£=i A i e i , where e n denotes the sequence with all co-ordinates 0, except for the nth co-ordinate which is 1. In this case we may assume without loss of generality that A is a Banach lattice, the ordering here being the usual co-ordinate- wise ordering. From the AK property it also follows that the dual A' of A may be identified with the a-dual (Ktithe dual) A a : = {ji: 1, i A i /i i < oo for all XeA}. Since A is symmetric, we may assume without loss of generality that ||ej| = 1. For information on scalar sequence spaces, refer to [6] and [9]. In Section 2 we introduce the concepts of A-Mackey convergence, A-Mackey convergence property and AS-family. These are generalizations of Mackey conver-