No abstract

We study the dependence of the Banach-Mazur distance between two subspaces of vector-valued continuous functions on the scattered structure of their boundaries. In the spirit of a result of Gordon [16], we show that the constant 2 appearing in the Amir-Cambern theorem may be replaced by 3 for some class of subspaces. This we achieve by showing that the Banach-Mazur distance of two function spaces is at least 3, if the height of the set of weak peak points of one of the spaces is larger than the height of a closed boundary of the second space. Next we show that this estimate can be improved, if the considered heights are finite and significantly different. As a corollary, we obtain new results even for the case of C(K, E) spaces.

No abstract

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