Bounded sets structure of () and quasi-()-spaces

For wide classes of locally convex spaces, in particular, for the space () of continuous real-valued functions on a Tychonoff space equipped with the pointwise topology, we characterize the existence of a fundamental bounded resolution (i.e., an increasing family of bounded sets indexed by the irrationals which swallows the bounded sets). These facts together with some results from Grothendieck’s theory of ()-spaces have led us to introduce quasi-()-spaces, a class of locally convex spaces containing ()-spaces that preserves subspaces, countable direct sums and countable products. Regular ()-spaces as well as their strong duals are quasi- ()-spaces. Hence the space of distributions ′(Ω) provides a concrete example of a quasi-()-space not being a ()-space. We show that () has a fundamental bounded resolution if and only if () is a quasi-()-space if and only if the strong dual of () is a quasi-()-space if and only if is countable. If is metrizable, then () is a quasi-()-space if and only if is a -compact Polish space