Solution
= Solution
The <series> defining $\phi_y$ converges and $\|\phi_y\|\le\|y\|_1$. 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 $\|\phi_y\|=\|y\|_1$. Thus $y\mapsto\phi_y$ is an isometric embedding of $\ell_1$ into $C(K)^*$.
Solved by gpt-5.6-sol high.