On linear homeomorphisms of spaces of continuous functions on «long lines»

In this paper, we prove that for the elementary regular ordinal and arbitrary ordinals $\alpha$, $\beta$, $\alpha<\beta\leqslant \tau$, the spaces of continuous functions $C_p(L_{\tau\cdot\alpha})$ and $C_p(L_{\tau\cdot\beta})$, defined on the "long lines" $L_{\tau\cdot\alpha}$ and $L_{\tau\cdot\beta}$, are not linearly homeomorphic.