Multi-normed spaces

Abstract

We modify the very well known theory of normed spaces (E, ‖ · ‖) within functional analysis by considering a sequence (‖ · ‖n : n ∈ N) of norms, where ‖ · ‖n is defined on the product space En for each n ∈ N. Our theory is analogous to, but distinct from, an existing theory of ‘operator spaces’; it is designed to relate to general spaces Lp for p ∈ [1,∞], and in particular to L1-spaces, rather than to L2-spaces. After recalling some results in functional analysis, especially in Banach space, Hilbert space, Banach algebra, and Banach lattice theory that we shall use, we shall present in Chapter 2 our axiomatic definition of a ‘multinormed space’ ((En, ‖ · ‖n) : n ∈ N), where (E, ‖ · ‖) is a normed space. Several different, equivalent characterizations of multi-normed spaces are given, some involving the theory of tensor products; key examples of multinorms are the minimum and maximum multi-norm based on a given space. Multi-norms measure ‘geometrically features’ of normed spaces, in particular by considering their ‘rate of growth’. There is a strong connection between multi-normed spaces and the theory of absolutely summing operators. A substantial number of examples of multi-norms will be presented. Following standard presentations of the foundations of functional analysis, we consider generalizations to ‘multi-topological linear spaces’ through ‘multi-null sequences’, and to ‘multi-bounded’ linear operators, which are exactly the ‘multi-continuous’ operators. We define a new Banach space M(E,F ) of multi-bounded operators, and show that it generalizes wellknown spaces, especially in the theory of Banach lattices. We conclude with a theory of ‘orthogonal decompositions’ of a normed space with respect to a multi-norm, and apply this to construct a ‘multidual’ space. Applications of this theory will be presented elsewhere. 2000 Mathematics Subject Classification. Primary 43A10, 43A20; secondary 46J10.