Spaces of \(N\)-homogeneous polynomials between Fréchet spaces. (English) Zbl 1073.46038
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.
Reviewer: Heinz Junek (Potsdam)
MSC:
| 46G25 | (Spaces of) multilinear mappings, polynomials |
| 46E50 | Spaces of differentiable or holomorphic functions on infinite-dimensional spaces |
| 46G20 | Infinite-dimensional holomorphy |
| 46A63 | Topological invariants ((DN), (\(\Omega\)), etc.) for locally convex spaces |
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