Köthe duals and Mackey topologies for nonclassical Lorentz spaces (Q789676)

Some results are presented for the nonclassical Lorentz space \(\Lambda_ G(\Omega)\) defined by a family \(G=\{G_{\omega}:\omega \in \Omega \}\) of concave maps. The measure space \(\Omega\) is taken to be either the unit interval or the natural numbers. It is shown that there is a measurable function g, computed from the \(G_{\omega}'s\), that determine the Köthe dual and Mackey topology for \(\Lambda_ G(\Omega)\). Specifically when \(\Omega\) is the unit interval the Köthe dual is given by \(\Lambda^*\!_ G=\{y:\sup_{s}\int^{s}_{0}\hat y(\omega)d\omega /g(s)<\infty \}\) where \(\hat y\) denotes the decreasing rearrangement of \(| y|\). In the case when the derivative of g is an element of \(\Lambda^*\!_ G\), the Mackey topology on \(\Lambda_ G\) coincides with the relative topology which \(\Lambda_ G\) inherits as a subspace of \(\Lambda_ G\!^{**}\). Under these circumstance, \(\Lambda_ G\) is locally convex if and only if \(\Lambda_ G=\Lambda_ G\!^{**}\). All of these results also hold for the corresponding sequence space \(\lambda_ G\).