  In this paper we consider a norm based on the infinitesimal generator of the
shift semigroup in a direction. The relevance of such a focus is guaranteed by
an abstract representation of a fractional integro-differential operator by
means of a composition of the corresponding infinitesimal generator. The main
result of the paper is a theorem establishing equivalence of norms in
functional spaces. Even without mentioning the relevance of this result for the
constructed theory, we claim it deserves to be considered itself.
