Duality of projective limit spaces and inductive limit spaces over a nonspherically complete non-Archimedean field

Abstract

A duality theorem of projective and inductive limit spaces over a nonspherically complete valued field is obtained under a certain condition, and topologies of spaces of locally analytic functions are ?studied. Introduction. Morita obtained in |5| a duality theorem of projec­ tive limit spaces and inductive limit spaces over a spherically complete nonarchimedean valued field, and Schikhof studied in [8J locally convex spaces over a nonsphorically complete nonarchimedean valued field. In this paper, we use the results of [8j and study the duality of such spaces over a nonspherically complete nonarchimedean valued field. The duality theorem of |5| was obtained as a generalization of the results of Komatsu |8| by Morita using: the theory of van Tiel [10] about locally convex spaces over a spherically complete nonarchimedean valued field. There the following two facts are used essentially: (i) The Mackey topology is the topology of uniform convergence on weakly e-compact sets; (ii) Any absolutely convex weakly c-compact set is strongly closed. Though we can generalize the notion of e-compactness to our ease, it is difficult to obtain good analogues of these two facts over a nonspherically complete valued field. Hence we restrict our attention to a more restricted class than in |5], and prove a duality theorem by making use of van dor Put’s duality theorem of sequence spaces c0 = {(au cl, aa, • • •')} and V {(&„ L, b„ ■••) X m (m Yn (m n(xn), y j m = (xn, vn_m{ym))n holds for any xn e X a and ym e Ym. Let (X, um) be the locally convex projective limit of {Xm, and let (Y, vm) be the locally convex inductive limit of [Ym, vnim}. We assume further that (iv) the projection map uw: X —> Xm has a dense image for each m. By definition, any element x of the projective limit X can be written as x — (xm) with xm e Xm satisfying umi„(x„) — xm for any m and n with m K by (x, y) = (uM(x), ym)m with such a ym 6 Ym. It is easy to see that this pairing ( , ) is iT-bilinear. Since the projection map um: X —> Xm is con­ tinuous, our pairing ( , ) is bicontinuous on X x Y m for each m. Hence, by the universal mapping property of the inductive limit topology, ( , ) is bicontinuous on X x Y . Let x = (xm) be a nonzero element of X Then xm =£ 0 for some m. Since ( , )m is nondegenerate, (xm, ym)m ^ 0 for some ym 6 Yn. Hence (*, y) = (xm, y j m # 0 for some y = vm(ym) e vm( Y J c Y. Let y = vm( y j (■ym e F J be a nonzero element of Y. Then {xm e Xm; (xm, ym)m ^ 0} is a non-empty open subset of Xm. Since the image of the projection map um: X -* X m is dense, there is an element x = (xm) e X such that (x, y) = (xm, ym)m ¥= 0. Therefore our pairing ( , ) is nondegenerate. Hence we have proved the following: DUALITY OVER A COMPLETE NONARCHIMEDEAN FIELD 389 P r o p o s i t i o n 1. Let X = proj lim and Y = ind lim Y m be as before* Then we have a nondegenerate bicontinuous K'-bilinear fo rm ( , ) : X x Y —> K .