We study the optimal approximation of the solution of an operator equation A(u)=f by linear and different types of nonlinear mappings. In our earlier papers we only considered the error with respect to a certain Hs-norm where s was given by the operator since we assumed that A:H0s(Ω)→H−s(Ω) is an isomorphism. The most typical case here is s=1. It is well known that for certain regular problems the order of convergence is improved if one takes the L2-norm. In this paper we study error bounds with respect to such a weaker norm, i.e., we assume that H0s(Ω) is continuously embedded into a space X and we measure the error in the norm of X. A major example is X=L2(Ω) or X=Hr(Ω) with r<s. We prove this better rate of convergence also for nonregular problems.