  We modify the very well known theory of normed spaces $(E, \norm)$ within
functional analysis by considering a sequence $(\norm_n : n\in\N)$ of norms,
where $\norm_n$ is defined on the product space $E^n$ for each $n\in\N$.
  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' $((E^n, \norm_n) : n\in \N)$, where $(E, \norm)$ 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.
