## Multi-norms and Banach lattices

### Volume 524 / 2017

#### Abstract

In 2012, Dales and Polyakov introduced the concepts of multi-norms and dual multi-norms based on a Banach space. Particular examples are the lattice multi-norm $(\|\cdot\|^L_n)$ and the dual lattice multi-norm $(\|\cdot\|^{DL}_n)$ based on a Banach lattice. Here we extend these notions to cover ‘$p$-multi-norms’ for $1\leq p\leq \infty$, where $\infty$-multi-norms and $1$-multi-norms correspond to multi-norms and dual multi-norms, respectively. We shall prove two representation theorems. First we modify a theorem of Pisier to show that an arbitrary multi-normed space can be represented as $((Y^n, \|\cdot\|_n^L) : n\in\mathbb N)$, where $Y$ is a closed subspace of a Banach lattice; we then give a version for certain $p$-multi-norms. Second, we obtain a dual version of this result, showing that an arbitrary dual multi-normed space can be represented as $(((X/Y)^n, \|\cdot\|_n^{DL}) : n\in\mathbb N)$, where $Y$ is a closed subspace of a Banach lattice $X$; again we give a version for certain $p$-multi-norms.

We shall discuss several examples of $p$-multi-norms, including the weak $p$-summing norm and its dual and the canonical lattice $p$-multi-norm based on a Banach lattice. We shall determine the Banach spaces $E$ such that the $p$-sum power-norm based on $E$ is a $p$-multi-norm. This relies on a famous theorem of Kwapień; we shall present a simplified proof of this result. We shall relate $p$-multi-normed spaces to certain tensor products.

Our representation theorems depend on the notion of ‘strong’ $p$-multi-norms, and we shall define these and discuss when $p$-multi-norms and strong $p$-multi-norms pass to subspaces, quotients, and duals; we shall also consider whether these multi-norms are preserved when we interpolate between couples of $p$-multi-normed spaces. We shall discuss multi-bounded operators between $p$-multi-normed spaces, and identify the classes of these spaces in some cases, in particular for spaces of operators between Banach lattices taken with their canonical lattice $p$-multi-norms.