The Ascoli property for function spaces

The paper deals with Ascoli spaces Cp(X) and Ck(X) over Tychonoff spaces X. The class of Ascoli spaces X, i.e. spaces X for which any compact subset K of Ck(X) is evenly continuous, essentially includes the class of kR-spaces. First we prove that if Cp(X) is Ascoli, then it is κ-Fréchet-Urysohn. If X is cosmic, then Cp(X) is Ascoli iff it is κ-Fréchet-Urysohn. This leads to the following extension of a result of Morishita: If for a Čech-complete space X the space Cp(X) is Ascoli, then X is scattered. If X is scattered and stratifiable, then Cp(X) is an Ascoli space. Consequently: (a) If X is a complete metrizable space, then Cp(X) is Ascoli iff X is scattered. (b) If X is a Čech-complete Lindelöf space, then Cp(X) is Ascoli iff X is scattered iff Cp(X) is Fréchet-Urysohn. Moreover, we prove that for a paracompact space X of point-countable type the following conditions are equivalent: (i) X is locally compact. (ii) Ck(X) is a kR-space. (iii) Ck(X) is an Ascoli space. The Ascoli spaces Ck(X,I) are also studied.