range of continous linear operator between banach spaces with closed algebraic complement

Question

Let X, Y be Banach spaces, and let A ∈ L(X, Y ). Suppose that there exists a closed subspace W ⊂ Y so that Y is the algebraic direct sum of ran A = A(X) and W. (That is, every y ∈ Y can be written as y = Ax + w for unique Ax ∈ ran A and w ∈ W.) I need to show that ran A is closed.

Answer 1

The natural linear isomorphism between Banach spaces $$(X/\ker A)\times W\to Y,\quad (x\bmod{\ker A},y)\mapsto A(x)+y$$ is continuous hence bicontinuous, and sends the closed subspace $(X/\ker A)\times\{0\}$ onto $\operatorname{ran}(A).$