Fernández, Antonio; Florencio, Miguel The space of essentially bounded measurable functions with values in a DF-space. (English) Zbl 0790.46021 Proc. R. Ir. Acad., Sect. A 93, No. 1, 87-95 (1993). Let \((\Omega,\Sigma,\mu)\) be a Radon measure space and \(E\) be a Hausdorff locally convex space. We study some properties of \(L^ \infty(\mu,E)\), the space of all essentially bounded and \(\mu\)-measurable functions from \(\Omega\) into \(E\) endowed with the topology of uniform convergence in \(\Omega\). In particular, we prove that \(L^ \infty(\mu,E)\) is a DF-space if and only if \(E\) is a DF-space and, in this case, we characterise the quasibarrelledness and barrelledness of the space \(L^ \infty (\mu,E)\). The main theorem of the paper is the followingTheorem. Let \((\Omega,\Sigma,\mu)\) be a measure space and \(E\) a DF- space.(a) If the measure \(\mu\) has atoms the following assertions are equivalent:(i) \(E\) is barrelled and each bounded subset of \(E\) is metrisable;(ii) \(L^ \infty(\mu,E)\) is barrelled.(b) If the measure \(\mu\) is atomless the following assertions are equivalent:(i) Each bounded subset of \(E\) is metrisable;(ii) \(L^ \infty(\mu,E)\) is barrelled. Reviewer: A.Fernández, M.Florencio (Sevilla) MSC: 46E40 Spaces of vector- and operator-valued functions 46A04 Locally convex Fréchet spaces and (DF)-spaces 46A08 Barrelled spaces, bornological spaces 46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) Keywords:Radon measure space; Hausdorff locally convex space; space of all essentially bounded and \(\mu\)-measurable functions; topology of uniform convergence; DF-space; quasibarrelledness; barrelledness PDFBibTeX XMLCite \textit{A. Fernández} and \textit{M. Florencio}, Proc. R. Ir. Acad., Sect. A 93, No. 1, 87--95 (1993; Zbl 0790.46021)