 Solution

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

W = ⟨ 1 ⟩

a \mapsto a 1

is an isometry from

R

W

S : Y \to X

(S g) (x) = \int_{0}^{x} g (t) d t .

S g \in C^{1} ([- 1, 1])

and

TS g = g

. Also,

∣ (S g) (x) ∣ \leq ∣ x ∣∥ g ∥_{\infty} \leq ∥ g ∥_{\infty}, ∥ S g ∥_{X} \leq ∥ g ∥_{Y} .

Thus

S

is a bounded right inverse, and

Y = C ([- 1, 1])

is a Banach space. Applying part (c),

(C^{1} ([- 1, 1]), ∥ f ∥_{\infty} + ∥ f^{'} ∥_{\infty}) is complete .

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