Isomorphisms between Spaces of Vector-valued Continuous Functions

A theorem due to Milutin [12] (see also [13]) asserts that for any two uncountable compact metric spaces Qt and Q2> t n e spaces of continuous real-valued functions C ^ ) and C(Q2) are linearly isomorphic. It immediately follows from consideration of tensor products that if X is any Banach space then QQ^X) and C(Q2;X) are isomorphic. The purpose of this paper is to show that this conclusion is false for general nonlocally convex quasi-Banach spaces. In fact, it fails in a quite strong manner. We shall show that if X is a quasi-Banach space containing no copy of c0 which is isomorphic to a closed subspace of a space with a basis and C(/;X) = C(A;I), (where / is the unit interval and A is the Cantor set) then we can conclude that X is locally convex. The proof requires building some machinery concerning operators on spaces of continuous functions. The locally convex analogues of these results are to be found in the work of Batt and Berg [1] or Brooks and Lewis [2]. Operators on spaces C(ft) into general non-locally convex spaces have been treated in an important paper of Thomas [19]. Unfortunately this paper has not been published. Thomas's main result can be expressed in the language of this paper as follows