Function spaces

AbstractFor a completely regular space X and a normed space E let Ck(X, E) (respectively Cp(X, E)) be the set of all E-valued continuous maps on X endowed with the compact-open (respectively pointwise convergence) topology. We prove that some topological properties P satisfy the following conditions: 1.(1) if Ck(X, E) and Ck(Y, F) (respectively Cp(X, E) and Cp(Y, F)) are linearly homeomorphic, then X ϵ P if and only if Y ϵ P;2.(2) if there is a continuous linear surjection from Ck(X, E) onto Cp(Y, F), then Y ϵ P provided X ϵ P;3.(3) if there is a continuous linear injection from Ck(X, E) into Cp(Y, F), then X has a dense subset with the property P provided Y has a dense subset with the same property