= Solution
Write $B a=\sum_{j=1}^n a_jN_j$, and let $R f=(f(x_1),\ldots,f(x_n))^T$. We use the standard interpolation convention of distinct ordered sites $x_1<\cdots<x_n$. Together with $N_i(x_i)>0$, the <Schoenberg–Whitney theorem> makes the <B-spline collocation matrix> $A$ invertible. Equivalently, its invertibility is implicit in the existence of the interpolation operator for every data vector.
If distinctness is not understood, the printed positivity condition alone is insufficient. For the order-two basis on <spline knots> $1,2,3,4$, take $x_1=x_2=5/2$. Both diagonal <B-spline> values are $1/2>0$, but the two rows of $A$ coincide. Arbitrary data cannot then be interpolated uniquely. The norm statement below concerns the intended well-defined interpolation operator.
<Spline interpolation> determines the coefficient vector by $Aa=Rf$, and hence
$$
P_{\mathbf x}=B A^{-1}R.
$$
Nonnegativity and the <subpartition of unity for B-splines> give
$$
\|Ba\|_\infty\le\|a\|_{\ell^\infty}.
$$
For completeness, extend the finite <spline knot sequence> beyond both ends. For the full sequence the order-one interval indicators sum to one. Summing the <Cox-de Boor recurrence> and shifting the index in its second term combines the two coefficients of each lower-order <B-spline> to one, so induction gives partition of unity at every order. Our finite collection is a subset of that nonnegative collection, and thus has sum at most one. Repeated <spline knots> are handled by the standard zero-term convention or a knot limit. This argument controls the entire interval, not only the basic knot interval.
Since $\|Rf\|_{\ell^\infty}\le\|f\|_\infty$, the <operator norm> upper bound is
$$
\|P_{\mathbf x}f\|_\infty
\le\|A^{-1}\|_{\ell^\infty}\|f\|_\infty.
$$
Here the matrix <operator norm> is the maximum absolute row sum. For the lower bound, choose a row of $A^{-1}$ with maximum absolute row sum and a data vector $y$ whose components are the signs of that row's entries. Then $\|y\|_{\ell^\infty}=1$ and
$$
\|A^{-1}y\|_{\ell^\infty}=\|A^{-1}\|_{\ell^\infty}.
$$
A continuous piecewise-linear function taking these values at the distinct ordered sites, and constant outside their range, has <supremum norm> one. Apply the given <uniform-norm stability of a B-spline basis> to its coefficient vector:
$$
\|P_{\mathbf x}f\|_\infty
=\|B A^{-1}y\|_\infty
\ge\frac1{d_k}\|A^{-1}y\|_{\ell^\infty}.
$$
Therefore
$$
\boxed{\frac1{d_k}\|A^{-1}\|_{\ell^\infty}
\le\|P_{\mathbf x}\|_{L^\infty}
\le\|A^{-1}\|_{\ell^\infty}}.
$$
The construction of the bounded continuous data extension is what permits the matrix norm to give a lower bound for a function-space <operator norm>.
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