A Note on the definition of an Orlicz space

Abstract

In the spaces \(L^p_\mu[0,1]\), \(p\gt 1\) (which are Orlicz spaces of a very special kind), one has \(\| f\|_p=\sup\{\int^1_0fg\,d\mu :\int^1_0|g|^q\,d\mu \leq1\}=\inf\{t\gt 0 :\int^1_0|f/t|^p\,d\mu\leq1\}=(\int^1_0|f|^p\,d\mu)^{1/p}\), and any one of the three could serve as the definition for \(\| f\|_p\). For an arbitrary \(N\)-function, \(M\), the analogues of the first two of these formulations yield equivalent norms on the corresponding Orlicz space. On the other hand, the functional \(\rho_M\), defined on \(L^M_\mu[0,1]\) by \(\rho_M(f)=M^{-1}(\int^1_0M (f)\,d\mu)\), not only fails to be a norm in most cases, but frequently it is not even a reasonable approximation to the Orlicz norm.