= Solution
The <orthogonal projection> condition is $(f-s^*,N_i)=0$ for all $i$. Since $M_i=kN_i/(t_{i+k}-t_i)$, it is equivalently
$$
Ga=b,\qquad G_{ij}=(M_i,N_j),\qquad b_i=(M_i,f).
$$
The <mixed-normalization spline Gram matrix> is invertible: $G=D^{-1}H$, where $D_{ii}=(t_{i+k}-t_i)/k>0$ and $H_{ij}=(N_i,N_j)$ is the positive-definite ordinary <Gram matrix> of the <basis>. Nonnegativity and the <unit-integral normalization of a B-spline> give $|b_i|\le\|f\|_\infty\int_0^1M_i=\|f\|_\infty$. The <subpartition of unity for B-splines> gives, on all of $[0,1]$,
$$
|s^*(t)|\le\sum_j|a_j|N_j(t)\le\|a\|_{\ell^\infty}.
$$
Consequently
$$
\|P_Sf\|_\infty\le\|a\|_{\ell^\infty}\le\|G^{-1}\|_{\ell^\infty}\|b\|_{\ell^\infty}\le\|G^{-1}\|_{\ell^\infty}\|f\|_\infty,
$$
so
$$
\boxed{\|P_S\|_\infty\le\|G^{-1}\|_{\ell^\infty}}.
$$
This is the <maximum-norm bound for spline projection>. Using a subpartition matters when the basic knot interval is a proper subinterval of $[0,1]$. As in the paper's assertion that the image lies in $C[0,1]$, the <spline> space here must consist of <continuous functions>; order-one step splines or full-multiplicity interior knots do not have that property.
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