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Norm Hilbert spaces over G-modules with a convex base
Indagationes Mathematicae ( IF 0.6 ) Pub Date : 2020-05-01 , DOI: 10.1016/j.indag.2020.03.003
E. Olivos , H. Ochsenius

By analogy with the classical definition, a Norm Hilbert space $E$ is defined as a Banach space over a valued field $K$ in which each closed subspace has an orthocomplement. In the rank one case (that is, the value group as well as the set of norms of the space are contained in $[0, \infty)$), they were described by van Rooij in his classical book of 1978, but the name itself was introduced in 1999 by Ochsenius and Schikhof for the case of spaces with an infinite rank valuation. Here we shall also consider only value groups that are contained in $(\R^+,\cdot)$, yet we borrow from the infinite rank case the notion of a $G$-module for the set of norms of the space. Their structure allows for greater complexity than that of ordered subsets of $\R$. In this paper we describe a new class of Norm Hilbert spaces, those in which the $G$-module has a convex base. Their characteristics will be the focus of our study.

中文翻译:

具有凸底的 G 模上的范数希尔伯特空间

通过与经典定义的类比,Norm Hilbert 空间 $E$ 被定义为值域 $K$ 上的 Banach 空间,其中每个闭合子空间都有一个正交补码。在秩一的情况下(即空间的值组和范数集合都包含在 $[0, \infty)$ 中),van Rooij 在他 1978 年的经典著作中对其进行了描述,但是name 本身是由 Ochsenius 和 Schikhof 于 1999 年针对具有无限秩估值的空间引入的。在这里,我们还将只考虑包含在 $(\R^+,\cdot)$ 中的值组,但我们从无限秩情况借用 $G$-模的概念来表示空间范数的集合。它们的结构允许比 $\R$ 的有序子集更大的复杂性。在本文中,我们描述了一类新的 Norm Hilbert 空间,$G$-module 具有凸底的那些。他们的特点将是我们研究的重点。
更新日期:2020-05-01
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