ON 2-HOMOGENEOUS POLYNOMIALS ON SOME NONSTABLE BANACH AND FRECHET SPACES

Let F be a Banach or a nuclear Frechet space isomorphic to its square. Then P(F-2), the space of 2-homogeneous polynomials on F, is isomorph ic to the space of continuous linear operators L(F, F'), both of them endowed with the topology of uniform convergence on bounded sets. In t his note we prove that the isomorphism can fail if F is not stable by studying two kind of examples: First, for Banach spaces, we consider J ames spaces J(p) constructed with the l(p)-norm, with p > 2; second, w e treat nuclear power spaces of finite or infinite type. (C) 1997 Acad emic Press.