Unboundedness of derivative evaluation in the supremum norm
= Unboundedness of derivative evaluation in the supremum norm
{title2=$f\mapsto f'(x_0)$}
On a nondegenerate interval, point evaluation of the <derivative> is unbounded for the <supremum norm> on smooth functions. The functions $f_N(x)=\sin(N(x-x_0))$ have <supremum norm> at most one but $f_N'(x_0)=N$. Adding the <supremum norm> of the <derivative> makes this evaluation a <bounded linear functional>.