= Solution
Use the standard partition normalization of <B-splines>, not rescaling each spline to have supremum exactly one. Let $h_i=t_{i+k}-t_i>0$. Then $M_i=kN_i/h_i$. In the ordinary real <inner product>, the <mixed-normalization spline Gram matrix> is $G=D^{-1}H$, where $D=\operatorname{diag}(h_i/k)$ and $H_{ij}=(N_i,N_j)$. Since the $N_i$ form a basis, $H$ is positive definite; $D$ is invertible, so $G$ is invertible. The undeclared upper index $h$ in the printed matrix should be $n$.
We need the standard positivity and normalization properties with their justification. The <Cox-de Boor recurrence> is
$$
N_{i,k}(t)=\frac{t-t_i}{t_{i+k-1}-t_i}N_{i,k-1}(t)+\frac{t_{i+k}-t}{t_{i+k}-t_{i+1}}N_{i+1,k-1}(t),
$$
with zero-width terms interpreted as zero. It starts from nonnegative interval indicators. On each lower-order spline's support, its recurrence coefficients are nonnegative, so induction gives $N_i\geq0$. Extend the finite knots outside their endpoints to a full knot sequence. In the full sum, coefficients of each lower-order spline add to one, so the recurrence preserves <partition of unity>. Our finite collection is a subset of nonnegative <B-splines>; therefore $\sum_{i=1}^nN_i(t)\leq1$ everywhere. The <Marsden identity> gives equality on the basic knot interval.
The divided-difference definition also gives support in $[t_i,t_{i+k}]$: below the left endpoint the knot-data function is a <polynomial> of degree $k-1$, whose order-$k$ <divided difference> vanishes; above the right endpoint the knot data vanish. Choose $a<t_i$ and $b>t_{i+k}$. Integrating in $t$ and then taking the <divided difference> in its knot variable gives
$$
\int M_i(t)\,dt
=k[t_i,\ldots,t_{i+k}]\left(\frac{(\cdot-a)^k}{k}\right)=1,
$$
because an order-$k$ <divided difference> of a monic degree-$k$ <polynomial> is one. The support lies inside $[0,1]$, so $\int_0^1M_i=1$.
<Orthogonality> of $f-s^*$ to the spline space gives $Ga=r$, with $r_i=(M_i,f)$. Since $M_i\geq0$ and has unit integral, $|r_i|\leq\|f\|_\infty$. The induced matrix maximum norm is the absolute row-sum norm, so
$$
\|a\|_{\ell^\infty}\leq\|G^{-1}\|_{\ell^\infty}\|f\|_\infty,
\qquad
|s^*(t)|\leq\|a\|_{\ell^\infty}\sum_iN_i(t)\leq\|a\|_{\ell^\infty}.
$$
Taking the supremum over nonzero $f$ proves \b[the <maximum-norm bound for spline projection>]:
$$
\boxed{\|P_{\mathcal S}\|_\infty\leq\|G^{-1}\|_{\ell^\infty}.}
$$
The range is $\mathcal S$, so the printed phrase “onto $C[0,1]$” must mean into $C[0,1]$, and onto $\mathcal S$. Continuity of the range requires continuous <B-splines>, for example $k\geq2$ and internal multiplicities at most $k-1$. For order-one or discontinuous <B-splines>, the same bound is valid with codomain $L^\infty[0,1]$ rather than $C[0,1]$.
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