FUNCTION-SPACES OF COMPLETELY METRIZABLE-SPACES

Let X and Y be metric spaces and let phi: C(p)(X) --> C(p)(Y) (resp. phi: C(p)*(X) --> C(p)*(Y)) be a continuous linear surjection. We prove that Y is completely metrizable whenever X is. As a corollary we obtain that complete metrizability is preserved by l(p) (resp. l(p)*-equivalence) in the class of all metric spaces. This solves Problem 35 in [2] (raised by Arhangel'skii).