On the duals of \({\mathfrak L}_p\) spaces with \(0<p<1\) (Q1416121)

Given a measurable space \(\langle \Omega, \mathfrak M, \mu\rangle\), a locally bounded Hausdorff topological linear space \((X,\tau)\) and a real number \(0<p<1\), one can define the space \(L_p(\Omega,\mathfrak M, \mu,X)\) which is, under certain assumptions, a Frèchet space for a suitable topology. \textit{M. M. Day} [Bull. Am. Math. Soc. 46, 816-823 (1940; Zbl 0024.21101)] gave necessary and sufficient conditions, in terms of properties of \(\langle \Omega, \mathfrak M,\mu\rangle\) for the dual of \(L_p(\Omega,\mathfrak M,\mu,\mathbb C)\) to be trivial. In this paper , a different proof for this result along with a slight generalization is given. The main result states that the dual of \(L_p\), in some sense, depends only on \(\langle \Omega,\mathfrak M,\mu\rangle\) being complete or not: The dual space of \(L_p\) is trivial if the measure space is complete. In addition, if \(\langle X,\tau\rangle\) is locally convex, the fact that the dual of \(L_p\) is trivial implies that \(\langle\Omega,\mathfrak M,\mu\rangle\) is complete.