On Dunford and Gelfand integrals in locally convex spaces (Q1194634)

The object of this paper is to study the Dunford and Gelfand integrability of a function defined on a complete finite measure space \((\Omega,\Sigma,\mu)\) with values in locally convex spaces. The authors have given some characterizations for the Dunford integrability and the equicontinuity of the range of the Dunford integral of a function defined on \(\Omega\) with values in a quasicomplete locally convex space \(X\) in terms of a linear operator associated with the function. Making use of this result, they have shown the following: if \((X^*,\beta(X^*,X))\) has the (GDF) property, then every scalarly integrable function \(f: \Omega\to X\) is Dunford integrable and the range of its Dunford integral is equicontinuous in \(X^{**}\) (the topological dual of \(X^*\) equipped with the strong topology \(\beta(X^*,X))\), and further, they have obtained that if \(f\) is bounded, it is always Dunford integrable and the range of its Dunford integral is equicontinuous in \(X^{**}\). Finally, they have defind the Gelfand integral for functions with values in the dual of a locally convex space and have obtained some results analogous to those for the Dunford integral.