Assume both X X and Y Y are Riemann surfaces which are subsets of compact Riemann surfaces X 1 X_1 and Y 1 , Y_1, respectively, and that the set X 1 − X X_1 - X has infinitely many points. We show that the only surjective complex linear isometries between the spaces of integrable holomorphic quadratic differentials on X X and Y Y are the ones induced by conformal homeomorphisms and complex constants of modulus 1. It follows that every biholomorphic map from the Teichmüller space of X X onto the Teichmüller space of Y Y is induced by some quasiconformal map of X X onto Y Y . Consequently we can find an uncountable set of Riemann surfaces whose Teichmüller spaces are not biholomorphically equivalent.