Norm of a two-endpoint interpolation projection ($\Vert P\Vert =\Vert |e_0|+|e_1|{\Vert }_{\infty }$)

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 $x_{\ast }$ maximizing $|e_0|+|e_1|$, choose unit complex numbers $a_j$ with $a_j{e}_j(x_{\ast })=|e_j(x_{\ast })|$. The continuous function $f(x)=(1-x)a_0+x{a}_1$ has supremum norm one and ${\ell }_j(f)=a_j$, so $(Pf)(x_{\ast })=|e_0(x_{\ast })|+|e_1(x_{\ast })|$. The L-infinity norm of the continuous output equals its supremum norm, proving equality even at an endpoint.