## Equivalence of multi-norms

### Volume 498 / 2014

#### Abstract

The theory of multi-norms was developed by H. G. Dales and M. E. Polyakov in a memoir that was published in *Dissertationes Mathematicae*.
In that memoir, the notion of `equivalence' of multi-norms was defined. In the present memoir, we make a systematic study of when various pairs of
multi-norms are mutually equivalent.

In particular, we study when $(p,q)$-multi-norms defined on spaces $L^r(\Omega)$ are equivalent, resolving most cases; we have stronger results in the case where $r=2$. We also show that the standard $[t]$-multi-norm defined on $L^r(\Omega)$ is not equivalent to a $(p,q)$-multi-norm in most cases, leaving some cases open. We discuss the equivalence of the Hilbert space multi-norm, the $(p,q)$-multi-norm, and the maximum multi-norm based on a Hilbert space. We calculate the value of some constants that arise.

Several results depend on the classical theory of $(q,p)$-summing operators.