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 E n 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 L p for p ∈ [1, ∞], and in particular to L 1 -spaces, rather than to L 2 -spaces.After recalling in Chapter 1 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 'multi-normed space' ((E n , · 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 multi-norms are the minimum, maximum, and (p, q)-multi-norms based on a given space. Multi-norms measure 'geometrical 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 the pattern of 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 well-known 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 'multi-dual' space.Applications of this theory will be presented elsewhere.

Ordered Banach algebras and multi-norms: some open problems

“…The notion of a p-multi-norm was first given in [17], under the name "type-p multinorm". They are mentioned in [9], whilst [8] is a memoir on this topic. The canonical p-multi-norm, ( · L ,P n ), on a Banach lattice E, for p ∈ [1, ∞), is defined by…”

“…The reader is referred to [9] for the primary definitions concerning multi-norms and to [5][6][7] for further material.…”

“…The theory of decompositions of Banach spaces in relation to multinorms involves orthogonal and small decompositions, using the language of §7.1 of [9]. Each small decomposition is orthogonal.…”

“…Problem 36 In [9], there is a duality theory for multi-norms that gives a multi-norm (not a dual multi-norm). Is there a similar duality theory for p-multi-norms that gives a p-multi-norm (not a p -multi-norm)?…”

Multi-norms

“…A theory of multi-norms based on a normed space E was first introduced by Dales and Polyakov in [7], and there have now been several papers on this topic. The present paper is a survey of the theory, somewhat expanded from the talk in Tartu.…”