= Norm of a two-endpoint interpolation projection
{title2=$\|P\|=\|\,|e_0|+|e_1|\,\|_\infty$}
Let $\ell_0,\ell_1$ be norm-one extensions to $L^\infty((0,1))$ of endpoint <point evaluations> on $C([0,1])$. For <continuous functions> $e_0,e_1$ with $e_i(j)=\delta_{ij}$, the <linear projection> $Pf=\ell_0(f)e_0+\ell_1(f)e_1$ has the displayed <operator norm>, independently of which extensions are chosen.
The <triangle inequality> proves the upper bound. At a point $x_*$ maximizing $|e_0|+|e_1|$, choose unit complex numbers $a_j$ with $a_je_j(x_*)=|e_j(x_*)|$. The <continuous function> $f(x)=(1-x)a_0+xa_1$ has <supremum norm> one and $\ell_j(f)=a_j$, so $(Pf)(x_*)=|e_0(x_*)|+|e_1(x_*)|$. The <L-infinity norm> of the continuous output equals its <supremum norm>, proving equality even at an endpoint.
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