= 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.
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