We modify the very well known theory of normed spaces $(E, orm)$ within functional analysis by considering a sequence $( orm_n : ninN)$ of norms, where $ orm_n$ is defined on the product space $E^n$ for each $ninN$. 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 $pin [1,infty]$, 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 $((E^n, orm_n) : nin N)$, where $(E, orm)$ 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.
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