 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/iii/paper-9/4/a/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

⟨ T_{n}, φ ⟩ \to ⟨ T, φ ⟩

φ \in C_{c}^{\infty} (Ω)

⟨ \partial_{x_{j}} T, φ ⟩ = - ⟨ T, \partial_{x_{j}} φ ⟩ .

It is linear and continuous: differentiation preserves compact support and bounds each test-function seminorm by one of higher derivative order, so composing it with the continuous functional

T

again gives a distribution. Finally, for every fixed test function,

⟨ \partial_{x_{j}} T_{n}, φ ⟩ = - ⟨ T_{n}, \partial_{x_{j}} φ ⟩ ⟶ - ⟨ T, \partial_{x_{j}} φ ⟩ = ⟨ \partial_{x_{j}} T, φ ⟩ .

Thus distributional differentiation preserves the stated convergence.

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