Spaces of \(N\)-homogeneous polynomials between Fréchet spaces (Q1883360)

The author studies properties of the Nachbin ported topology \(\tau_{\omega}\) on the space \({\mathcal P}(^NE,F)\) of \(N\)-homogeneous polynomials between Fréchet spaces \(E\) and \(F\). To achieve this, the canonical linear topological isomorphism \(({\mathcal P}(^NE,F),\tau_\omega) \cong L_\omega(\bigotimes _{N,s,\pi}E,F)\) is used. Here, the topological space \(L_\omega(G,F)\) is defined as \(L_\omega(G,F)=\text{proj}_n \text{ind}_m L_b(G_m,F_n)\), where \((G_m)\) and \((F_n)\) denote the canonical spectra of \(G\) and \(F\), respectively. This linearization allows to study the barrelledness of \(({\mathcal P}(^NE,F),\tau_\omega)\) in terms of \(\text{Ext}^1(\widehat{\bigotimes}_{N,\pi}E,F)=0\). Using certain permanence properties of Vogt's \((DN)\) and \((\overline{\overline{\Omega}})\) properties for LB-spaces, some necessary and sufficient criteria for the barrelledness of \({\mathcal P}(^NE,F),\tau_\omega)\) are given.