# Pre-composing a Closed operator by a bounded operator

A linear map ( not necessarily bounded ) between normed linear spaces is called a closed operator if its graph is closed. Suppose $$X$$ is a n.l.s and $$Y, Z$$ are Banach spaces. Let $$A : X_0 ⊆ X → Y$$ be a closed operator and $$B ∈ B(Z, X)$$ such that $$Ran(B) ⊆ X_0$$ . Prove that $$AB ∈ B(Z, Y )$$

Since, $$Z,Y$$ both are given to be Banach, I was willing to apply Closed graph Theorem to $$AB$$ . Let $$(x,y)$$ be a limit point of the Graph, then Enough to show that $$AB(x)=y$$ . Consider any sequence $$\{(x_n,ABx_n\}_{n \ge 1}$$ in the Graph such that it converges to $$(x,y)$$ i.e. $$x_n \to x$$ and $$ABx_n \to y$$ . Since , B is given to be continuous, $$x_n \to x \implies B(x_n) \to B(x)= z$$ (say) .

So all we need to show is $$Az=y$$ . Here I am stuck. How to use the fact that $$A$$ admits a closed graph?

Note that $$(Bx_n, ABx_n)$$ is a sequence in the graph of $$A$$. You know $$Bx_n \to Bx$$ and $$ABx_n \to y$$ so, since the graph of A is closed, $$Bx \in D(A)$$ and $$ABx =y$$. This implies that $$x \in D(AB)$$ and $$ABx =y$$ which is what we needed to show that $$AB$$ has closed graph.