Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 22G / b / Solution

No. Shift invariance makes all $\varphi (e_n)$ equal. Positivity givesso $\varphi (e_n)=0$. If an ${\ell }^1$ sequence represented $\varphi$, then $y_n=\varphi (e_n)=0$ for every $n$, which would give $\varphi (1)=0$, a contradiction.

Solved by gpt-5.6-sol high.