= Solution
This addresses the general interpolation bound in part (I); the subsequent (a)–(c) address the cubic case in part (II). Write the interpolant in the <B-spline> <basis> as
$$
P_{\mathbf x}f=\sum_{j=1}^nc_jN_j,\qquad A_{\mathbf x}c=Rf,\qquad Rf=(f(x_1),\ldots,f(x_n))^T.
$$
The stipulated uniquely defined <spline interpolation operator> presupposes invertibility of the <B-spline collocation matrix>. For the standard increasing distinct sites, the conditions $N_i(x_i)>0$ give it by the <Schoenberg–Whitney theorem>. Thus $c=A_{\mathbf x}^{-1}Rf$.
Use the standard partition-normalized <B-splines> of the question: they are nonnegative and satisfy the <subpartition of unity for B-splines>, $\sum_jN_j(t)\leq1$. This also holds outside the basic knot interval; one may extend the knots and regard the finite family as a subfamily of a full partition of unity. The recurrence proof of the full partition is given in Question 6(c). Repeated knots retain the same nonnegativity and subpartition properties, with zero-denominator recurrence terms interpreted as zero or by knot limits. Consequently
$$
\left|\sum_jc_jN_j(t)\right|\leq\sum_j|c_j|N_j(t)\leq\|c\|_{\ell^\infty}.
$$
Sampling has <norm> at most one, $\|Rf\|_{\ell^\infty}\leq\|f\|_\infty$, and so
$$
\|P_{\mathbf x}f\|_\infty\leq\|A_{\mathbf x}^{-1}\|_{\ell^\infty}\|f\|_\infty,
\qquad\boxed{\|P_{\mathbf x}\|_{L^\infty}\leq\|A_{\mathbf x}^{-1}\|_{\ell^\infty}.}
$$
Here the induced <matrix> <norm> is the maximum absolute row sum. The label $L^\infty$-normalized means the usual partition normalization, not rescaling each <spline> to have maximum value one. Such a rescaling would invalidate the specific <coefficient> bound and the cardinal values used below.
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