  An old question of A.V. Arhangel'skii asks if the Menger property of a
Tychonoff space $X$ is preserved by homeomorphisms of its function space
$C_p(X)$. We provide affirmative answer in the case of linear homeomorphisms.
To this end, we develop a method of studying invariants of linear
homeomorphisms of fuction spaces $C_p(X)$ by looking at the way $X$ is
positioned in its (\v{C}ech-Stone) compactification.
