We examine the topological properties of Orlicz-Bochner spaces \(L^\varphi(X)\) (over a σ-finite measure space \((\Omega, \Sigma, \mu))\), where \(\varphi\) is an Orlicz function (not necessarily convex) and \(X\) is a real Banach space. We continue the study of some class of locally convex topologies on \(L^\varphi (X)\), called uniformly \(\mu\)-continuous topologies. In particular, the generalized mixed topology \(\mathcal{T}_I^\varphi (X)\) on \(L^\varphi (X)\) (in the sense of Turpin) is considered.
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Let E be a Banach function space and let X be a real Banach space. We examine weakly compact linear operators from a Köthe-Bochner space E(X) endowed with some natural mixed topology $γ_{E(X)}$ (in the sense of Wiweger) to a Banach space Y.
Let \(E\) be an ideal of \(L^0\) over \(\sigma\)-finite measure space \((\Omega, \Sigma, \mu)\) and let \((X, \| \cdot \|_X)\) be a real Banach space. Let \(E(X)\) be a subspace of the space \(L^0(X)\) of \(\mu\)-equivalence classes of all strongly \(\Sigma\)-measurable functions \(f\colon \Omega \to X\) and consisting of all those \(f\in L^0(X)\), for which the scalar function \(\tilde{f} = \|f (\cdot)\|_X\) belongs to \(E\). Let \(E\) be equipped with a Hausdorff locally convex-solid topology \(\xi\) and let \(\xi\) stand for the topology on \(E(X)\) associated with \(\xi\). We examine the relationship between the properties of the space \((E(X), \xi)\) and the properties of both the spaces \((E, \xi)\) and \((X, \|· \|_X)\). In particular, it is proved that \(E(X)\) (embedded in a natural way) is an order closed ideal of its bidual iff \(E\) is an order closed ideal of its bidual and \(X\) is reflexive. As an application, we obtain that \(E(X)\) is perfect iff \(E\) is perfect and \(X\) is reflexive.
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