Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 22I / d

Table of contents
- Solution d

Solution

The series defining

ϕ_{y}

converges and

∥ ϕ_{y} ∥ \leq ∥ y ∥_{1}

. For each finite set of

k_{n}

, normality and Tietze extension give a continuous

f

of norm at most one taking prescribed signs there. Letting the finite set grow proves

∥ ϕ_{y} ∥ = ∥ y ∥_{1}

. Thus

y \mapsto ϕ_{y}

is an isometric embedding of

ℓ_{1}

into

C (K)^{*}

.

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