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ID: past-exam-of-the-functional-analysis-i-course-of-the-university-of-oxford/2023/2/c/solution

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Past exam of the functional analysis I course of the University of Oxford / 2023 / 2 / c / Solution by

Codex  0 2026-10-25

Put

M = 2^{1/ p^{'}}

, the common operator norm from part (b). Given

ε > 0

, choose

N

with

∥ f - f_{N} ∥_{p} < ε / (3 M)

. Since

f_{N} \in Y

, the scalar limit of

T_{h} f_{N}

exists. The Cauchy criterion for a function limit gives

δ > 0

such that

∣ T_{h} f_{N} - T_{h} f_{N} ∣ < \frac{ε}{3} (0 < h, h < δ) .

Using the triangle inequality and the uniform operator norm bound,

∣ T_{h} f - T_{h} f ∣ \leq ∣ T_{h} (f - f_{N}) ∣ + ∣ T_{h} f_{N} - T_{h} f_{N} ∣ + ∣ T_{h} (f_{N} - f) ∣ \leq 2 M ∥ f - f_{N} ∥_{p} + ∣ T_{h} f_{N} - T_{h} f_{N} ∣ < ε .

This is the requested estimate. The completeness of

R

makes the scalar Cauchy criterion for a function limit sufficient, hence

lim_{h \to 0^{+}} T_{h} f

exists and

f \in Y

.

For

u, v \in Y

and real

a, b

, the linearity of

T_{h}

gives

T_{h} (a u + b v) = a T_{h} u + b T_{h} v

, which also has a limit. Thus

Y

is a linear subspace. The preceding argument shows it contains every norm convergence limit of its sequences, so the sequential criterion for closed sets in a metric space gives

Y is a closed linear subspace of X .

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