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 $\Vert g+\psi {\Vert }_{\infty }\ge |g(x_0)|+1$ 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 $C([0,1])\oplus ⟨\psi ⟩$, prescribe $L_a(g+c\psi )=g(x_0)+ca$ for $|a|\le 1$. Scaling the estimate gives $\Vert g+c\psi {\Vert }_{\infty }\ge |g(x_0)|+|c|\ge |L_a(g+c\psi )|$, 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 $\Vert \psi {\Vert }_{\infty }=1$.