 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-2/22h/b/solution

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

The identity map

I : (V, ∥ \cdot ∥_{1}) \to (V, ∥ \cdot ∥_{2})

is a continuous linear bijection. Since both spaces are Banach, the bounded inverse theorem makes

I^{- 1}

continuous, giving

∥ x ∥_{1} \leq C_{2} ∥ x ∥_{2}

Completeness is essential. On the space

c_{00}

of finitely supported sequences,

∥ x ∥_{2} \leq ∥ x ∥_{1}

, but for

x = (1, \dots, 1, 0, \dots)

∥ x ∥_{1} /∥ x ∥_{2} = n

is unbounded. Both normed spaces are incomplete.

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