Multi-normed spaces

We modify the very well known theory of normed spaces $(E,\text{norm})$ within functional analysis by considering a sequence $({\text{norm}}_n:n\in \mathbb{N})$ of norms, where ${\text{norm}}_n$ is defined on the product space $E^n$ for each $n\in \mathbb{N}$. 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 $p\in [1,\infty]$, 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’ $((E^n,{\text{norm}}_n):n\in \mathbb{N})$, where $(E,\text{norm})$ 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 and maximum multi-norm 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 $\mathcal{M}(E,F)$ 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.