Lattice-Normed Spaces

Abstract

In this chapter we consider structural properties of a vector space with some norm taking values in a vector lattice. Such a vector space is called a lattice-normed space; an LNS for short. The most important peculiarities of LNSs are connected with the decomposability property (2.1.1 (4)). The latter allows us, in particular, to indicate a complete Boolean algebra of linear projections in a lattice-normed space, which is isomorphic and closely related to the Boolean algebra of band projections of the norm lattice (2.1.3, 2.1.4). Moreover, a decomposable LNS admits a compatible module structure over a certain ring of orthomorphisms (2.1.8). These facts are closely related with the disjointness relation induced by the vector norm (2.1.2). If an LNS is simultaneously a vector lattice then there is another disjointness relation connected with its lattice structure. Some simple interrelation between them is reflected in the notions of norm-indecomposable and norm-n-decomposable elements (2.1.9). It turns out that every norm-n-decomposable elements is very often the sum of n norm-indecomposable elements (2.1.10). Partitions of unity in a Boolean algebra lead to the operation of mixing elements in a lattice-normed space (2.2.1). If there exists a mixing of every (vector) norm-bounded family in a lattice-normed space then the space is called disjointly complete (2.1.5). For instance, such are Banach-Kantorovich spaces (BKSs), i.e. decomposable and order complete lattice-normed spaces (2.2.1).