 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-1/22g/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

No. Shift invariance makes all

ϕ (e_{n})

equal. Positivity gives

0 \leq Nϕ (e_{1}) = ϕ (e_{1} + \dots + e_{N}) \leq ϕ (1) = 1,

ϕ (e_{n}) = 0

. If an

ℓ^{1}

sequence represented

ϕ

, then

y_{n} = ϕ (e_{n}) = 0

for every

n

, which would give

ϕ (1) = 0

, a contradiction.

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