Bounded sets structure of C p X and quasi‐ ( D F ) ‐spaces

For wide classes of locally convex spaces, in particular, for the spaceC p ( 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 ( D F ) ‐spaces have led us to introduce quasi‐ ( D F ) ‐spaces, a class of locally convex spaces containing ( D F ) ‐spaces that preserves subspaces, countable direct sums and countable products. Regular ( L M ) ‐spaces as well as their strong duals are quasi‐ ( D F ) ‐spaces. Hence the space of distributionsD ′ ( Ω )provides a concrete example of a quasi‐ ( D F ) ‐space not being a ( D F ) ‐space. We show thatC p ( X )has a fundamental bounded resolution if and only ifC p ( X )is a quasi‐ ( D F ) ‐space if and only if the strong dual ofC p ( X )is a quasi‐ ( D F ) ‐space if and only if X is countable. If X is metrizable, thenC k ( X )is a quasi‐ ( D F ) ‐space if and only if X is a σ‐compact Polish space.