THE WEAK AND STRONG CONVERGENCE IN METRIZABLE SPACES OVER VALUED FIELDS

Abstract: Its is proved that a metrizable tvs E over a spherically complete non-archimedean non-trivially valued ¯eld is non-archimedean i ® every weak null-sequence in E is a null-sequence. Throughout this paper IK = (IK; j j) denotes a spherically complete non-archimedean non-trivially complete valued ¯eld. A topological vector space (tvs) (E; ¿) over IK is said to be locally convex (or F-convex as in [6]) if ¿ has a basis of absolutely convex neighbourhoods of zero, where a subset A of E is called absolutely convex (in the sense of Monna [5]) if ax + by 2 A provided x; y 2 A, a; b 2 IK, jaj · 1, jbj · 1. Normed (F-normed) spaces E are de¯ned in a natural way; a norm (F-norm) k k on E is called non-archimedean (n.a.) if kx+yk · maxfkxk; kykg, x; y 2 E. A normed (F-normed) space E is called non-archimedean (n.a.) if the original topology of E can be de¯ned by a n.a. norm (n.a. F-norm). Clearly every n.a. normed (F-normed) space is locally convex; by Proposition III.4 of [1] every normed space which as a tvs is locally convex must be n.a.; similarly one proves the same for F-normed spaces. In [5], p. 74, Monna discussed relations between the convergence and the weak convergence of sequences in some normed spaces. It is known that every weakly convergent sequence in a locally convex space (E; ¿) over a spherically complete ¯eld IK is ¿-convergent; this follows directly from Proposition 4.3 of [2] and Theorem 4.12 of [6]. In [4], Theorem 3, Martinez-Maurica and Perez-Garcia improved some results of Monna by showing that a normed space E over a spherically complete ¯eld IK is n.a. i ® every weakly convergent sequence in E is convergent. The assumption concerning the spherical completness (for the de¯nition see [5]) can not be removed, cf. [7], Remark, p. 200