On the "three-space problem" for spaces of polynomials.

Abstract

A property P of locally convex spaces is called a three-space property whenever the following implication holds: if both a closed subspace F and the corresponding quotient E/F of a locally convex space E have P then E has P as well. The authors consider properties P of the form: E has P whenever two "natural'' topologies coincide on the spaces of n-homogeneous polynomials on E. They consider topologies of the uniform convergence on all absolutely convex compact or bounded subsets as well as the strong topology and the Nachbin ported topology. The results obtained are mostly negative and the counterexamples are variations of the known spaces.