AI 中文总结
本文研究巴拿赫格中序半连续性的变体性质，建立其关系与重赋范稳定性，证明序连续赋范格的刻画、半连续AM-空间的刻画等结论，统一简化单调完备性刻画的证明并给出定量版本。
AI 中文摘要
本文研究赋范格的多种性质，这些性质与半连续性（又称Fatou性质）类似，建立这些性质之间的关系，并讨论它们在重赋范下的稳定性。特别地，我们证明赋范格F是序连续的当且仅当F的每一次重赋范都是半连续的。我们还证明弱Fatou赋范格（我们提出术语“半连续的”）恰好是同构于单调完备巴拿赫格的正则子格的那些格。为了做到这一点，我们引入半连续赋范格的Lorentz完备化概念，它在某种程度上类似于向量格理论中的通用完备化。此外，我们统一并简化了来自文献[aw]和[Taylor]的单调完备性刻画的证明，并提供了它们的定量版本。AM-空间中与半连续性相关的性质具有一些额外特征。经典Kakutani定理指出AM-空间恰好是Co(K)-空间的闭子格，我们使用两种不同方法证明半连续AM-空间恰好是Co(K)-空间的闭正则子格。最后，我们证明对于赋范空间E，Ba_{E*}上的正齐次弱*连续函数的AM-空间是半连续的当且仅当它是Co(Ba_{E*},w*)的正则子格当且仅当dim E<∞。
英文摘要
In this article we consider various properties of a normed lattice, which are similar to semicontinuity (also known as the Fatou property), establish relations between these properties, and discuss stability of these properties under renorming. In particular, we show that a normed lattice $F$ is order continuous iff every renorming of $F$ is semicontinuous. We also prove that the weakly Fatou (we propose the term ``demicontinuous'') normed lattices are precisely the ones isomorphic to regular sublattices of monotonically complete Banach lattices. In order to do so we introduce the concept of the Lorentz completion of a demicontinuous normed lattice, which is somewhat analogous to the universal completion from the vector lattice theory. Furthermore, we unify and simplify the proofs of the characterizations of monotone completeness from \cite{aw} and \cite{taylor} and provide their quantitative versions. The semicontinuity-related properties in AM-spaces have some additional features. While the classical Kakutani theorem states that AM-spaces are precisely the closed sublattices of $\Co\left(K\right)$-spaces, we show, using two different methods, that the semicontinuous AM-spaces are precisely the closed regular sublattices of $\Co\left(K\right)$-spaces. Finally, we prove that for a normed space $E$, the AM-space of positively homogeneous weak* continuous functions on $\Ba_{E^{*}}$ is semicontinuous iff it is a regular sublattice of $\Co\left(\Ba_{E^{*}},\mathrm{w}^{*}\right)$ iff $\dim E<\8$.
Comments37 pages, preliminary version
