Completeness of a graph norm from a bounded right inverse ($\ker T\text{ complete},TS=I,S\text{ bounded}⟹(X,\Vert \cdot {\Vert }_T)\text{ complete}$)

Let $X$ be a normed space, $Y$ a Banach space, and $T:X\to Y$ a linear map. If its kernel is complete in the original norm and $S:Y\to X$ is a bounded right inverse, then the graph norm $\Vert x{\Vert }_T=\Vert x{\Vert }_X+\Vert Tx{\Vert }_Y$ is complete, even if $X$ with its original norm is incomplete or $T$ is unbounded.

For a Cauchy sequence $(x_n)$ in the graph norm, $y_n=T{x}_n$ converges to $y\in Y$. The elements $w_n=x_n-S{y}_n$ lie in the kernel and satisfy $\Vert w_n-w_m{\Vert }_X\le (1+\Vert S\Vert )\Vert x_n-x_m{\Vert }_T$. Hence $w_n\to w\in \ker T$. The element $x=w+Sy$ satisfies $Tx=y$ and