Generalizing algebraically defined norms

We extend the algebraic construction of finite dimensional varying exponent L-p(.) space norms, defined in terms of Cauchy polynomials to a more general setting, including varying exponent L-p(.) spaces. This boils down to reformulating the Musielak-Orlicz or Nakano space norm in an algebraic fashion where the infimum appearing in the definition of the norm should become a (uniquely attained) minimum. The latter may easily fail, as turns out, and in this connection we examine the Fatou type semicontinuity conditions on the modulars. Norms defined by ODEs are applied in studying such semicontinuity properties of L-p(.) space norms with p(.) unbounded.