Function spaces

For 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. 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; there is a continuous linear surjection from Ck(X, E) onto Cp(Y, F), then Y ϵ P provided X ϵ P; 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.