Q-reflexive Locally Convex Spaces

Abstract

For a locally convex space $E$, we use the Aron–Berner extension to define canonical mappings from ${\hat{\otimes }}_{s,n,\pi }{E}^{\prime \prime }e$ into different duals of $\mathcal{P}({}^{n}E)$. We investigate necessary and suffcient conditions for the continuity of these mappings, paying particular attention to three special cases — Fréchet spaces, DF spaces and reflexive A-nuclear spaces. We define Q-reflexive spaces as spaces where a certain canonical mapping can be extended to an isomorphism between ${\hat{\otimes }}_{s,n,\pi }{E}^{\prime \prime }e$ and $\overline{(\mathcal{P}({}^{n}E),{\tau }_b{)}_i^{\prime }}$. We find examples of such spaces.