= Solution
The entries in (b) satisfy
$$
|g_{ii}|-\sum_{j\ne i}|g_{ij}|\ge\frac23-\frac{h_i+h_{i+1}}{3(h_i+h_{i+1})}=\frac13.
$$
Thus $G$ has <strict diagonal dominance> with margin at least $1/3$. To prove the <inverse infinity-norm bound from diagonal dominance> directly, choose $i$ with $|a_i|=\|a\|_{\ell^\infty}$. The <reverse triangle inequality> gives
$$
\|Ga\|_{\ell^\infty}\ge|(Ga)_i|\ge\left(|g_{ii}|-\sum_{j\ne i}|g_{ij}|\right)|a_i|\ge\frac13\|a\|_{\ell^\infty}.
$$
This also proves that the <kernel> is zero and hence that $G$ is invertible. Substituting $a=G^{-1}b$ and taking the <operator norm> gives
$$
\boxed{\|G^{-1}\|_{\ell^\infty}\le3}.
$$
Combined with (a), it yields $\|P_S\|_\infty\le3$ for linear <splines>, independently of the knot spacings. The diagonal-dominance argument already proves the required estimate.
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