The mesh independence principle states that, if Newton’s method is used to solve an equation on Banach spaces as well as finite dimensional discretizations of that equation, then the behaviour of the discretized process is essentially the same as that of the initial method. This principle was inagurated in Allgower et al. (SIAM J Numer Anal 23(1):160–169, 1986). Using our new Newton–Kantorovich-like theorem and under the same information we show how to extend the applicability of this principle in cases not possible before. The results can be used to provide more efficient programming methods.