On the strict topology in the non-locally convex setting

Abstract

Let X be a completely regular space and let E be a locally convex space. Denote by Crc(X, E) the space of all continuous E-valued functions on X with relatively compact range. The dual space of Crc(X, E) under the uniform topology is a space M(B, E') of U-valued measures (see [5]). In case E is a locally convex lattice, M(B, E') becomes a lattice. In [3] the author defined the topologies fl and fl, on C,c(X, E). The corresponding dual spaces are the spaces M~(B, E') and M~(B, E') of all z-additive and all a-additive members of M(B, E') respectively. In this paper we look into the problem of the [3(fl 0 equicontinuity of weakly compact subsets of the positive cone of M~(B, E') (M,,(B, E')) when E is a locally convex lattice. In the scalar case it is known that each such set is fl(flO equicontinuous (see [12]). It is shown here that this is not always true in the vector-valued case. More specifically it is shown that, for a completely regular space X, the following assertions are equivalent: (1) X is compact (pseudocompact). (2) For each Banach lattice E, fl(fll) is the topology of uniform convergence on the weakly compact subsets of the positive cone of Me(B, E') (M~(B, E')). In case E is finite dimensional, it is shown that (C~(X, E), ill) is a strong Mackey space and that (Crc(X, E), fl) is a strong Mackey space iff (Cb(X), fl) is a strong Mackey space [Cb(X) is the space of all bounded continuous real functions on X]. Finally it is proved that, for E a Banach space and ), = fl or//1 the space (C,c(X, E), 7) is a strong Mackey space iff it is Mackey.