Multi-normed spaces

We modify the very well known theory of normed spaces (E, parallel to . parallel to) within functional analysis by considering a sequence (parallel to . parallel to(n) : n is an element of N) of norms, where parallel to . parallel to(n) is defined on the product space E-n for each n is an element of 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 is an element of [1, infinity], 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-formed space' ((E-n, parallel to . parallel to(n)) : n is an element of N), where (E, parallel to . parallel to) 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.