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The space of essentially bounded measurable functions with values in a DF-space. (English) Zbl 0790.46021

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 following
Theorem. 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.

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.)
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