The topology of almost uniform convergence

Abstract

set S into a locally convex linear topological space F. Then a subset U of &(S, F) has property β over a subset A of S if it satisfies the following condition: for some neighborhood V of 0 in F it is true that for each finite subset {f19 ° ,/fc} of %>*(S, F) ~ U there is a finite subset {xlf x2, , xn] of A and a finite set of positive numbers {alf a2, •••, an}, Σ?=i α i = 1> s u c h that Σϊ-i^/X^*) * s n ° t ' V for j — 1,2, ...,fc. 5.4 THEOREM. Consider the function space %?(S, F). Then all sets of the form Ux Π Π Un, where each U,h has property (β) over some A in s/, form a local base for a locally convex topology. This is called the topology of convex almost uniform convergence on members of sf. Furthermore, it is a linear topology if and only if f [A] is bounded for each A in Sf and each f in Z?(S, F) and it is a Hausdorff topology if F is Hausdorff and for each f in 2Γ(S, F) there is a point x in at least one member of ,S>f such that f(x) Φ 0. The omitted proof of the above theorem is essentially the same as Theorem 2.4. 5.5 THEOREM. Let S be a linear topological space and let ^ 7 ( S , F) be a collection of continuous linear functions defined on S with range in a locally convex linear topological space F. If s/ is a family of subsets of S such that £?{S, F) is a linear topological space for the topology of convex almost uniform convergence on members of s^f and if j y is the collection of closed convex hulls of finite unions of members of S?/, then the topology of almost uniform convergence on the members of S/ is the same topology. Proof. In collaboration with Theorem 4.1 it is sufficient to show that a subset U of jSf (S, F) has property (β) over a subset A of S if and only if it has property (a) on the convex hulls of A. Because of the linearity of the members of U the result becomes apparent upon inspecting Definitions 5.3 and 1.2,