Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 22H / b / Solution

The identity mapis a continuous linear bijection. Since both spaces are Banach, the bounded inverse theorem makes $I^{-1}$ continuous, giving $\Vert x{\Vert }_1\le C_2\Vert x{\Vert }_2$.

Completeness is essential. On the space $c_{00}$ of finitely supported sequences, $\Vert x{\Vert }_2\le \Vert x{\Vert }_1$, but for $x=(1,\dots ,1,0,\dots )$ the ratio $\Vert x{\Vert }_1/\Vert x{\Vert }_2=\sqrt{n}$ is unbounded. Both normed spaces are incomplete.

Solved by gpt-5.6-sol high.