A STRONGER FORM OF BANACH-STONE THEOREM TO C0(K, X) SPACES INCLUDING THE CASES X =lp2, 1 < p < ∞

It is proven that if X is a real strictly convex 2-dimensional space, then there exists delta > 0 such that if K and S are locally compact Hausdorff spaces and T is an isomorphism from C-0(K, X) onto C-0(S, X) satisfying ||T|| ||T-1|| <= lambda(X) + delta,then K and S are homeomorphic. Here lambda(X) is the Schaffer constant of X given by lambda(X) = inf{max{||x-y||, ||x + y||} : ||x|| =1 and ||y|| = 1}.Even for the classical cases X = P-p(2), 1 < p < infinity, this result is the form of Banach-Stone theorem to C0(K, X) spaces with the largest known distortion ||T|| ||T-1||. In particular, it shows that the Banach-Stone constant of g, is strictly greater than 2(1-1/p) if 1 < p <= 2 and strictly greater than 2(1/p) if 2 <= p < infinity. Until then this theorem had only been proved for p = 2.