A strict topology for some weighted spaces of continuous functions

Abstract

In the classical case the strict topology $\beta$ introduced by Buck [2] on the space $C^b(X)$ of bounded continuous scalar valued functions on the locally compact Hausdorff space X is given by the system W of all weights on X that vanish at infinity. The $\beta$-bounded subsets of $C^b(X)$ are exactly the norm bounded subsets, and $\beta$ is the finest locally convex topology which coincides on the norm bounded subsets with the compact open topology (cf. Dorroh [4]). Especially we have that $C^b (X) = CW(X) = CW_0 (X)$ holds algebraically. In this note we want to describe for an arbitrary system of weights V an associated system of weights W such that at least in many cases, including the classical one, the connection between $CV(X)$ and $CW_0 (X)$ is the same as in the classical case.