Diffeological spaces are generalizations of smooth manifolds which include singular spaces and function spaces. For each diffeological space, Iglesias-Zemmour introduced a natural topology called the D-topology. However, the D-topology has not yet been studied seriously in the existing literature. In this paper, we develop the basic theory of the D-topology for diffeological spaces. We explain that the topological spaces that arise as the D-topology of a diffeological space are exactly the ∆-generated spaces and give results and examples which help to determine when a space is ∆-generated. Our most substantial results show how the D-topology on the function space C ∞ (M, N ) between smooth manifolds compares to other well-known topologies.

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SynopsisNecessary and sufficient conditions on weights u and v are given for the inequalityto hold for all g e C%(R").

SynopsisA necessary and sufficient condition for the boundedness of the generalised Stieltjes transformation on weighted Lebesgue spaces is obtained and applied to extend the Hilbert double series theorem.