Dieudonn\'{e} completeness of function spaces

A space is called Dieudonn\'{e} complete if it is complete relative to the maximal uniform structure compatible with its topology. In this paper, we investigated when the function space $C(X,Y)$ of all continuous functions from a topological space $X$ into a uniform space $Y$ with the topology of uniform convergence on a family of subsets of $X$ is Dieudonn\'{e} complete. Also we proved a generalization of the Eberlein-\v{S}mulian theorem to the class of Banach spaces.