Let $X,Y,Z$ Banach spaces, $\text{dom}(S)\subset Y$, let $T:X\rightarrow Y$ be linear and continuous and let $S:\text{dom}(S)\rightarrow Z$ be linear and closed. Show that the composition $ST$ is also closed.
I think the open mapping theorem might be applicable, but I don't know weather $\text{dom}(S)\cap\text{Im}(T)$ is closed.