论文标题
直接限制规范矢量晶格和Banach Lattices
Direct limits in categories of normed vector lattices and Banach lattices
论文作者
论文摘要
在按间隔和几乎间隔保存线性图和矢量晶格同态的几乎间隔收集结果后，我们表明，各种规范矢量晶格和Banach晶格中的直接系统具有直接的限制，并且这些系统与自然相关的其他类别中系统的直接限制一致。对于那些一般构造无法确定一般直接限制的存在的类别，我们描述了确实存在的那些直接限制的基本结构。 Banach晶格和一定程度的几乎间隔保存矢量晶格同态的直接系统具有直接的限制。当系统中的Banach晶格都具有连续规范时，Banach晶格也可以直接限制。这用于表明当局部紧凑的Hausdorff空间上的Banach函数空间具有连续范围，当所有紧凑型子集上的拓扑都可以进行计量，并且（（连续紧凑的）函数的图像是密集的。
After collecting a number of results on interval and almost interval preserving linear maps and vector lattice homomorphisms, we show that direct systems in various categories of normed vector lattices and Banach lattices have direct limits, and that these coincide with direct limits of the systems in naturally associated other categories. For those categories where the general constructions do not work to establish the existence of general direct limits, we describe the basic structure of those direct limits that do exist. A direct system in the category of Banach lattices and contractive almost interval preserving vector lattice homomorphisms has a direct limit. When the Banach lattices in the system all have order continuous norms, then so does the Banach lattice in a direct limit. This is used to show that a Banach function space over a locally compact Hausdorff space has an order continuous norm when the topologies on all compact subsets are metrisable and (the images of) the continuous compactly supported functions are dense.
