A strict dual representation for \(L_ 1(\mu{},X)\) (Q1173801)

For a given complete finite measure space \((\Omega,\Sigma,\mu)\) the authors consider the Banach space \(X\) and the strict topologies \(\beta_ 1\), \(\beta_ 2\) on \(C(S,X^*_ \tau)\), where \(S\) is a Stone space of the measure algebra \(\Sigma/\mu^{-1}(0)\) and \(X^*\) is taken with Mackey topology. The most important result is the idea of using a differentiability criterion to represent \(L_ 1(\mu,X)\) as the dual of an appropriate space of continuous functions with an appropriate strict topology (\(L_ 1(\mu,X)\) is the space of Bochner integrable \(X\)-valued functions on \(\Omega\)). The last part of this paper is devoted to characterize the subsets of \(L_ 1(\mu,X)\) which are equicontinuous with respect to \(\beta_ 1\) and \(\beta_ 2\). The paper also contains very valuable and important examples.