Bounded sets structure of CpX and quasi‐(DF)‐spaces

For wide classes of locally convex spaces, in particular, for the space Cp(X) of continuous real‐valued functions on a Tychonoff space X 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 (DF)‐spaces have led us to introduce quasi‐(DF)‐spaces, a class of locally convex spaces containing (DF)‐spaces that preserves subspaces, countable direct sums and countable products. Regular (LM)‐spaces as well as their strong duals are quasi‐(DF)‐spaces. Hence the space of distributions D′(Ω) provides a concrete example of a quasi‐(DF)‐space not being a (DF)‐space. We show that Cp(X) has a fundamental bounded resolution if and only if Cp(X) is a quasi‐(DF)‐space if and only if the strong dual of Cp(X) is a quasi‐(DF)‐space if and only if X is countable. If X is metrizable, then Ck(X) is a quasi‐(DF)‐space if and only if X is a σ‐compact Polish space.