= Nonuniqueness of norm-preserving point evaluation extensions
View $C([0,1])$ inside the complex <Lp space> $L^\infty((0,1))$. Fix $x_0\in[0,1]$. If a unit <function> $\psi$ satisfies $\|g+\psi\|_\infty\geq|g(x_0)|+1$ for every <continuous function> $g$, then the possible values at $\psi$ of norm-one extensions of <point evaluation> at $x_0$ form exactly the closed complex unit disc.
The estimate first excludes $\psi\in C([0,1])$. On the <direct sum> $C([0,1])\oplus\langle\psi\rangle$, prescribe $L_a(g+c\psi)=g(x_0)+ca$ for $|a|\leq1$. Scaling the estimate gives $\|g+c\psi\|_\infty\geq|g(x_0)|+|c|\geq|L_a(g+c\psi)|$, and $L_a(1)=1$ proves <operator norm> one. The <complex Hahn-Banach theorem> extends $L_a$ to the whole <Lp space>. No value outside the disc is possible because $\|\psi\|_\infty=1$.
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