 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-4/22h/c/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

The upper frame bound gives

∥ Ux ∥_{ℓ^{2}} \leq B ∥ x ∥

. For

c \in ℓ^{2}

, the adjoint is the norm-convergent synthesis series

U^{*} c = \sum_{i} c_{i} e_{i},

because its inner product with

x

\sum_{i} c_{i} ⟨ e_{i}, x ⟩

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