On 2-Homogeneous Polynomials on Some Non-stable Banach and Fr´echet Spaces

Let F be a Banach or a nuclear Fr´echet space isomorphic to its square. Then P 2F., the space of 2-homogeneous polynomials on F, is isomorphic to the space of continuous linear operators L F, F9., both of them endowed with the topology of uniform convergence on bounded sets. In this 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 James spaces Jpconstructed with the lp-norm, with p)2; second, we treat nuclear power spaces of finite or infinite type.