In this paper, some criteria for weakly approximative compactness and approximative compactness of weak$ ^{*} $ hyperplane for Musielak-Orlicz-Bochner function spaces are given. Moreover, we also prove that, in Musielak-Orlicz-Bochner function spaces generated by strongly smooth Banach space, $ L_{M}^{0}(X) $ (resp $ L_{M}(X) $) is an Asplund space if and only if $ M $ and $ N $ satisfy condition $ \Delta $. As a corollary, we obtain that $ L_{M}^{0}(R) $ (resp $ L_{M}(R) $) is an Asplund space if and only if $ M $ and $ N $ satisfy condition $ \Delta $.
Citation: Shaoqiang Shang, Yunan Cui. Weak approximative compactness of hyperplane and Asplund property in Musielak-Orlicz-Bochner function spaces[J]. Electronic Research Archive, 2020, 28(1): 327-346. doi: 10.3934/era.2020019
In this paper, some criteria for weakly approximative compactness and approximative compactness of weak$ ^{*} $ hyperplane for Musielak-Orlicz-Bochner function spaces are given. Moreover, we also prove that, in Musielak-Orlicz-Bochner function spaces generated by strongly smooth Banach space, $ L_{M}^{0}(X) $ (resp $ L_{M}(X) $) is an Asplund space if and only if $ M $ and $ N $ satisfy condition $ \Delta $. As a corollary, we obtain that $ L_{M}^{0}(R) $ (resp $ L_{M}(R) $) is an Asplund space if and only if $ M $ and $ N $ satisfy condition $ \Delta $.
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Shaoqiang Shang, Yunan Cui. Weak approximative compactness of hyperplane and Asplund property in Musielak-Orlicz-Bochner function spaces[J]. Electronic Research Archive, 2020, 28(1): 327-346. doi: 10.3934/era.2020019
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