 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-327/1/a/solution

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

S (R^{n}) = {φ \in C^{\infty} (R^{n}) : p_{α, β} (φ) < \infty for all multi-indices α, β},

where

p_{α, β} (φ) = sup_{x \in R^{n}} ∣ x^{α} D^{β} φ (x) ∣.

φ_{j}

converges to

φ

S

exactly when every one of these seminorms of

φ_{j} - φ

tends to zero.

S^{'} (R^{n})

is the continuous dual of

S (R^{n})

. Its standard weak convergence is

u_{j} ⟶ u ⟺ ⟨ u_{j}, φ ⟩ ⟶ ⟨ u, φ ⟩ for every φ \in S .

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