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

Take the normed space $X=C^1([-1,1])$ with the supremum norm, and let $T:X\to C([-1,1])$ be the inclusion $Tf=f$. This is an injective linear map, so its kernel is the complete zero space. Its graph norm isThe Cauchy sequence $f_n(x)=\sqrt{x^2+n^{-2}}$ from part (d)(ii) still has no limit in $X$ for this norm, because its only possible limit is $|x|$. Consequently