 Banach norm rigidity from continuous point evaluations

On $C(X)$ for compact Hausdorff $X$, any complete norm making every point evaluation functional continuous is equivalent to the uniform norm. If $f_n$ converges in the new norm to $f$ and uniformly to $g$, evaluation gives $f(x)=g(x)$ for every $x$, so the identity map between the two Banach spaces has closed graph. The closed graph theorem and open mapping theorem make the identity and its inverse bounded. Completeness of the proposed norm is essential to this argument.