= Solution
The <orthogonal projection> has range $\mathcal S$, not all of $C[0,1]$: the latter is an <infinite-dimensional vector space> whereas $\mathcal S$ is a <finite-dimensional vector space>. For a concrete counterexample, $e^x$ cannot belong to any finite-degree piecewise polynomial space, since all its <derivatives> are nonzero on every open interval. For $k\ge2$, the <B-splines> are <continuous>, so the <orthogonal projection> is a map $C[0,1]\to\mathcal S\subset C[0,1]$. For order one it instead has a generally discontinuous piecewise constant range. For example, a single interval indicator on $[1/4,3/4)$ is the projection of the constant <function> one onto its span, and is not <continuous>.
Write $s^*=\sum_ja_jN_j$. Since $M_i$ is a positive <scalar multiple> of $N_i$, the <operator normal equations> $\langle f-s^*,N_i\rangle=0$ are equivalent to
$$
Ga=b,\qquad b_i=\langle M_i,f\rangle.
$$
The <mixed-normalization spline Gram matrix> is invertible: $G=DH$ where $D_{ii}=k/(t_{i+k}-t_i)>0$ and $H_{ij}=\langle N_i,N_j\rangle$ is the ordinary <positive-definite matrix> of <inner products> of a linearly independent <basis>. The positivity and <unit-integral normalization of a B-spline> give $|b_i|\le\|f\|_\infty$. The <subpartition of unity for B-splines> gives
$$
|s^*(t)|\le\|a\|_{\ell^\infty}\sum_jN_j(t)\le\|a\|_{\ell^\infty}.
$$
Here the <operator norm> of a <matrix> on $\ell^\infty$ is its maximum absolute row sum. Consequently,
$$
\|P_{\mathcal S}f\|_\infty\le\|a\|_{\ell^\infty}\le\|G^{-1}\|_{\ell^\infty}\|b\|_{\ell^\infty}\le\|G^{-1}\|_{\ell^\infty}\|f\|_\infty,
\qquad\boxed{\|P_{\mathcal S}\|_\infty\le\|G^{-1}\|_{\ell^\infty}.}
$$
This proves the <maximum-norm bound for spline projection>.
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