Past exam of the functional analysis I course of the University of Oxford / 2023 / 1 / a

Both terms in the graph norm are finite and nonnegative. Moreover, $\Vert x{\Vert }_T=0$ forces $\Vert x{\Vert }_X=0$, hence $x=0$; conversely $T0=0$ gives $\Vert 0{\Vert }_T=0$. This proves definiteness of a norm.

The linearity of $T$ and absolute homogeneity of a norm give, for each real $a$,Applying the triangle inequality in both normed spaces givesThus