Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 22H / b

If $T$ is invertible, the bounded inverse theorem makes $T^{-1}$ bounded, andMoreover $T^{\ast }$ is invertible with inverse $(T^{-1}{)}^{\ast }$, so the same argument shows that $T^{\ast }$ is bounded below.

Conversely, supposefor some $c,d>0$. The first bound makes $T$ injective. It also makes $\mathrm{ran}T$ closed: if $T{x}_n$ is Cauchy, then $x_n$ is Cauchy, and continuity gives its limit in the range. The second bound gives $\ker T^{\ast }=\{0\}$. By image-kernel orthogonality for an adjoint,so the range is dense. Being both closed and dense, it equals $H$. Thus $T$ is bijective, and the first lower bound gives $\Vert T^{-1}\Vert \le c^{-1}$. Therefore