This work focuses on the {\em a posteriori} error estimation for the reduced basis method applied to partial differential equations with quadratic nonlinearity and affine parameter dependence. We rely on natural norms --- {\em local} parameter-dependent norms --- to provide a sharp and computable lower bound of the inf-sup constant. We prove a formulation of the Brezzi--Rappaz--Raviart existence and uniqueness theorem in the presence of two distinct norms. This allows us to relax the existence condition and to sharpen the field variable error bound. We also provide a robust algorithm to compute the Sobolev embedding constants involved in the error bound and in the inf-sup lower bound computation. We apply our method to a steady natural convection problem in a closed cavity, with Grashof number varying from 10 to $10^7$.