On a problem of topologies in infinite dimensional holomorphy (Q923328)

In 1988, \textit{J. Taskinen} [Stud. Math. 91, No.1, 17-30 (1988; Zbl 0654.46060)] constructed a Fréchet-Montel space F such that F\({\hat \otimes}_{\pi}F\) contains \(\ell^ 1\). Therefore, this tensor product cannot be a Montel space. Using this space F and the representation of the space \(P(^ 2F)\) of the 2-homogeneous polynomials as a subspace of \(F'{\hat \otimes}_{\epsilon}F'\) it is shown that \(P(^ 2F)\) cannot be a Montel space too. As a consequence, the compact open topology and the Nachbin topology differ on each space H(U) of holomorphic functions on any open balanced subset U of F.