Function Spaces and Dieudonn#xE9; Completeness

摘要

For a completely regular spaceXand a normed spaceEletCk(x, E) (resp.,Cp(x, E)) be the set of allE-valued continuous maps onXendowed with the compact-open (resp., pointwise convergence) topology. It is shown that the set of allF-valued linear continuous maps onCk(x, E) when equipped with the topology of uniform convergence on the members of some families of bounded subsets ofCk(x, E) is a complete uniform space ifFis a Band space andXis Dieudonn#xE9; complete. This result is applied to prove that Dieudonn#xE9; completeness is preserved by linear quotient surjections fromCk(x, E) ontoCk(Y, E) (resp., fromCp(x, E) ontoCp(x, E)) providedE,Fare Band spaces andYis ak-space.