 Solution

The entries in (b) satisfyThus $G$ has strict diagonal dominance with margin at least $1/3$. To prove the inverse infinity-norm bound from diagonal dominance directly, choose $i$ with $|a_i|=\Vert a{\Vert }_{{\ell }^{\infty }}$. The reverse triangle inequality givesThis also proves that the kernel is zero and hence that $G$ is invertible. Substituting $a=G^{-1}b$ and taking the operator norm givesCombined with (a), it yields $\Vert P_S{\Vert }_{\infty }\le 3$ for linear splines, independently of the knot spacings. The diagonal-dominance argument already proves the required estimate.