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Use the standard $L_{\infty }$-oriented partition normalization of B-splines, in contrast to the unit-integral normalization. Thus $N_i\ge 0$ and ${\sum }_i{N}_i(t)\le 1$ on $[0,1]$, with equality for a complete basis on its basic knot interval. This terminology does not mean that each basis function's maximum is exactly one; for example an order-three cardinal spline has maximum $3/4$. Assume the basis has positive support on the integration interval and is continuous, as is needed for a projection taking values in $C[0,1]$.

LetThe ordinary Gram matrix $H$ is positive definite by linear independence. The appropriate row-normalized, or mixed-normalization spline Gram matrix, isHere $M_i$ is a unit-integral nonnegative spline. Unlike $H$, $G$ need not be symmetric. If the full support lies inside the interval, $d_i=(t_{i+k}-t_i)/k$; otherwise use the actual interval integral rather than that full-support formula.

The normal equations for the $L_2$ orthogonal projection $P_{\mathcal{S}}f={\sum }_i{a}_i{N}_i$ say $Ha=c$, with $c_i=\int f{N}_i$. Divide each row by $d_i$ to obtainNonnegativity and unit integral give $|b_i|\le \Vert f{\Vert }_{\infty }$. The subpartition of unity for B-splines then givesTaking the supremum over $t$ and then over nonzero $f$ proves the uniform norm bound for B-spline orthogonal projection:The row normalization is essential; replacing $G$ with the ordinary Gram matrix would not be the same asserted estimate. The operator is defined on $C[0,1]$ and maps into its finite-dimensional spline subspace, onto $\mathcal{S}$, not onto all of $C[0,1]$ as one phrase in the PDF says. Its range follows both from the formula and from reproducing every $s\in \mathcal{S}$.

If “$L_{\infty }$-normalized” were instead read literally as individually peak-one splines, their sum can exceed one. The basis-dependent factor $K=\Vert {\sum }_i{N}_i{\Vert }_{\infty }$ must then be retained: $\Vert P_{\mathcal{S}}{\Vert }_{\infty }\le K\Vert (D^{-1}H{)}^{-1}{\Vert }_{{\ell }^{\infty }}$. Equivalently the displayed bound without an outside factor uses $G=K^{-1}{D}^{-1}H$. The standard partition-normalized convention has $K\le 1$ and yields the sharper bound proved above.