 Solution

ID: past-exam-of-the-functional-analysis-i-course-of-the-university-of-oxford/2023/1/e/solution

X = C^{1} ([- 1, 1])

with the supremum norm, and let

T : X \to C ([- 1, 1])

be the inclusion

T f = f

. This is an injective linear map, so its kernel is the complete zero space. Its graph norm is

∥ f ∥_{T} = 2∥ f ∥_{\infty} .

f_{n} (x) = x^{2} + n^{- 2}

from part (d)(ii) still has no limit in

X

for this norm, because its only possible limit is

∣ x ∣

. Consequently

ker T = {0} is complete, but (X, ∥ \cdot ∥_{T}) is incomplete .

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