Continuity of injective basis separating maps

We prove that certain (“basis separating”) linear injections are automatically continuous. We discuss openness of such maps in Section 5. There are two stages to the proof of continuity: (1) An injective basis separating map can be written in a canonical form (Theorem 4.3). (2) Any map of this form is continuous (Theorem 4.4). Given Banach spaces X and Y with Schauder bases {xn} and {yn}, respectively, we say that H:X→Y, H(∑n∈Nx(n)xn)=∑n∈NHx(n)yn, is basis separating if for all elements x=∑n∈Nx(n)xn and y=∑n∈Ny(n)xn of X, x(n)y(n)=0 for all n∈N implies that Hx(n)Hy(n)=0 for all n∈N. Associated with any linear basis separating map H, there is a support map h:N→N∞ that we discuss in Section 3. The support map enables us to develop the canonical form (Eq. (3.2)) for basis separating maps.