Equivalence of multi-norms

Volume 498 / 2014

H. G. Dales, M. Daws, H. L. Pham, P. Ramsden Dissertationes Mathematicae 498 (2014), 1-53 MSC: Primary 46B28; Secondary 46M05, 47L05. DOI: 10.4064/dm498-0-1


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.


  • H. G. DalesFylde College
    University of Lancaster
    Lancaster LA1 4YF
    United Kingdom
  • M. DawsSchool of Mathematics
    University of Leeds Leeds LS2 9JT, UK
  • H. L. PhamSchool of Mathematics, Statistics
    and Operations Research
    Victoria University of Wellington
    Wellington 6140, New Zealand
  • P. Ramsden5 Brookhill Crescent
    Leeds LS17 8QB, UK

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