Linear equivalence of (pseudo) compact spaces

Given Tychonoff spaces X and Y, Uspenskij proved in [15] that if X is pseudocompact and Cp(X) is uniformly homeomorphic to C_p(Y), then Y is also pseudocompact. In particular, if C_p(X) is linearly homeomorphic to C_p(Y), then X is pseudocompact if and only if so is Y. This easily implies Arhangel’skii’s theorem [1] which states that, in the case when C_p(X) is linearly homeomorphic to C_p(Y the space X is compact if and only if Y is compact. We will establish that existence of a linear homeomorphism between the spaces C_p*(X)  and C_p*(Y) implies that X is (pseudo)compact if and only if so is Y. We will also show that the methods of proof used by Arhangel’skii and Uspenskij do not work in our case.