Arch. Math., Vol. 60, 157-163 (1993) 0003-889X/93/6002-0157 $ 2.90/0 1993 Birkhfiuser Vertag, Basel On an analogue of Hardy's inequality By P. D. JOHNSONJR. and R. N. MOHAPATRA 1. Definitions and background. Let co be the vector space of all real or complex se- quences and q~ be the subspace of all eventually zero sequences, i.e. sequences with only finitely many non-zero entries. A sequence space is a subspace of co containing ~b. If (Pn) is a sequence of positive real numbers, we write (~.1) and t(po) = {x ~o;.:o ~; Ix"r"~ < o0} (1.2) ces(p,) = eco; [Xk[ < Oe . n=O k=O Ifp, = p for all n, then l(p,) = Ip, and ces (p,) = CeSp (see [7], [3], [4], and [8] [11].) If (p,) is unbounded, then l (p,) is closed neither under addition nor scalar multiplica- tion; we require (p,) to be bounded here, in which case l(p,) is a sequence space. Ifpn < M for all n, then l(p,)~= l~ and ces(p,)~ ces M. Therefore, if p, < 1 for all n, then ces (p,) __cces 1 = {0}. It is possible for ces (p,) to be non-trivial even though p, < 1 for infinitely many n (the set of indices n for which p, < 1 will be rather sparse in such cases), and it is possible for ces (p,) to be trivial even though p, > I for all n; nonetheless, not wishing to investigate the former phenomenon here, and bearing in mind the dangers of the latter, we shall require Pn > I throughout. With (N) thus confined, l(p,) is a complete metric topological sequence space, with topology defined in the usual way by the function IIxlJ~p.~ = Z IXnJP~ n=0 It is a consequence of the results in [3] [6] that ces(p,) is either trivial, or a sequence space, and with topology defined in the obvious way, enjoys every conventional property possessed by l(p,), including completeness. For 0 < p, < 1, Simons [11] has studied the properties of l(pn ) as a topological vector space. Maddox [9] has studied the K6the-Toeplitz and linear topological duals of the /(p,,). Maddox and Roles [10] have obtained results on k-convexity of the l (Pn) and other related spaces.