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## Commentationes Mathematicae

2008 | 48 | 1 |
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### Duality and some topological properties of vector-valued function spaces

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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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2008
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2017-12-19
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