= Uniform norm bound for B-spline orthogonal projection
{title2=$\|P_{\mathcal S}\|_\infty\le\|G^{-1}\|_{\ell^\infty}$}
Let nonnegative partition-normalized <B-splines> $N_i$ form a <basis>, with $\sum_iN_i\le1$ on the integration interval and $d_i=\int N_i>0$. The row-normalized mixed <Gram matrix> is $G_{ij}=\int N_iN_j/d_i$. The <normal equations> for the <orthogonal projection> onto their <linear span> become $Ga=b$, with $b_i=\int fN_i/d_i$ and $\|b\|_{\ell^\infty}\le\|f\|_\infty$. The <subpartition of unity for B-splines> gives $\|\sum_i a_iN_i\|_\infty\le\|a\|_{\ell^\infty}$ and proves the bound. The ordinary symmetric Gram matrix omits this essential row normalization. The projection maps continuous functions into the spline space, not onto all continuous functions.
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