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 $L^{r}(Ω)$ 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 $L^{r}(Ω)$ 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.