Isomorphisms of $${\mathcal {C}}(K, E)$$ Spaces and Height of K

Abstract Let $$K_1$$ K 1 , $$K_2$$ 2 be compact Hausdorff spaces and $$E_1, E_2$$ E , Banach not containing a copy of $$c_0$$ c 0 . We establish lower estimates the Banach–Mazur distance between continuous functions $${\mathcal {C}}(K_1, E_1)$$ C ( ) {C}}(K_2, E_2)$$ based on ordinals $$ht(K_1)$$ h t $$ht(K_2)$$ which are new even for case real-valued ordinal intervals. As corollary we deduce that isomorphic if is substantially different from