The Five Lemma for Banach Spaces

We prove a version of the five lemma which is useful for the study of boundary value problems for partial differential equations. The results are given in the category <!-- MATH $\mathcal{B}$ --> of Banach spaces and bounded linear operators, and all conditions are stated modulo an arbitrary ideal of <!-- MATH $\mathcal{B}$ --> . We also show that the results are valid in more general categories.