On topological and linear homeomorphisms of certain function spaces (Q1122809)

The main result of the paper is the following theorem: If X is a countable metric space which is not locally compact, then the function space \(C_ p(X)\) is homeomorphic to \(\sigma_{\omega}=(\ell^ 2_ f)^{\infty}\). A technical tool of the investigation of function spaces in the paper is the notion of Q-matrix due to J. van Mill. However the authors mentioned that the theorem follows from the result of \textit{T. Dobrowolski}, \textit{S. P. Gul'ko} and \textit{J. Mogilski} (submitted to Topology Appl.) on the topological equivalence \(C_ p(X)\approx \sigma_{\omega}\) for any countable metric nondiscrete space X. Two examples are given to show that function spaces \(C_ p(X)\) and \(C_ p(Y)\) need not be linearly homeomorphic for separable metric spaces X and Y which are not locally compact.