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Weakly sequentially complete Fréchet spaces of integrable functions. (English) Zbl 0921.46040

Summary: For a Fréchet space \(E\) let \(L^1(\mu,E)\) be the space of all \(E\)-valued integrable functions with respect to a probability measure \(\mu\) and, when \(E\) is a real Fréchet space, let \(L^1({\mathcal G},E)\) be the space of all real-valued integrable functions with respect to an \(E\)-valued vector measure \({\mathcal G}\). We prove that if \(E\) is weakly sequentially complete then both \(L^1(\mu,E)\) and \(L^1({\mathcal G},E)\) are weakly sequentially complete. We also prove that if \(E\) is a Fréchet lattice, then \(E\) is weakly sequentially complete if and only if \(E\) does not contain any (lattice) copy of \(c_0\). By combining these main theorems, we also give some results about the existence of copies of \(c_0\) in \(L^1(\mu,E)\) and \(L^1({\mathcal G},E)\).

MSC:

46G10 Vector-valued measures and integration
46A04 Locally convex Fréchet spaces and (DF)-spaces
46A40 Ordered topological linear spaces, vector lattices
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