Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 22I / b / iii / Solution

Let $X=c_{00}$, the finitely supported sequences, with the ${\ell }^2$ norm. This normed space is not complete. Let $Y={\ell }^2$ and defineBoth maps $X\to Y$ are bounded, and $S$ is injective. Moreoverbecause multiplication of a finitely supported sequence by $(1,2,3,\dots )$ preserves finite support. Butso $\Vert (S^{-1}T)e_n{\Vert }_2=n$ while $\Vert e_n{\Vert }_2=1$. Hence $S^{-1}T$ is not continuous, showing that completeness of $X$ is essential.

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