Function spaces
For a completely regular space \(X\) and a normed space \(E\) let \(C_k(X,E)\) (respectively \(C_p(X,E)\)) be the set of all continuous maps from \(X\) to \(E\) equipped with the compact-open (respectively pointwise convergence) topology. The paper studies properties which are preserved by linear continuous surjections from \(C_k(X,E)\) onto \(C_p(Y,F)\) or linear homeomorphisms between \(C_k(X,E)\) and \(C_k(Y,F)\), as well as between \(C_p(X,E)\) and \(C_p(Y,F)\). These properties include (under some restrictions on \(X\) and \(Y\)) compactness, Čech-completeness, local compactness, analyticity, K-analyticity and others. The result concerning Čech-completeness generalizes the one of Baars, de Groot and Pelant (Čech-completeness is preserved by continuous linear surjections from \(C_p(X)\) onto \(C_p(Y)\) in the class of metric spaces), the result concerning preservation of local compactness by \(l_p\)-equivalence generalizes the one proved by Gulko and Okunev, further by R. A. McCoy and I. Ntantu and also by Baars and de Groot. The author also answers a question of Baars and de Groot.
scientific article; zbMATH DE number 1088817
scientific article