For a locally convex space $E$ we use the Aron-Berner extension to define canonical mappings from $\pin E_e''$ into different duals of $\sP(^nE)$. We investigate necessary and sufficient 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 $\pin E_{e}''$ and $\overline{(\sP(^nE),\t_b)_{i}'}$. We find examples of such spaces.
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