= Solution
For the usual <spline interpolation> problem the sites are distinct and ordered; the <Schoenberg–Whitney theorem> then makes $A_x$ invertible under the stated support positivity. More generally, the following argument applies whenever the interpolation map in the question is well defined and $A_x$ is invertible.
Factor the <spline interpolation operator> as $P_x=BA_x^{-1}R$, where
$$
Rf=(f(x_1),\ldots,f(x_n)),\qquad Bc=\sum_{j=1}^nc_jN_j.
$$
Sampling has <operator norm> at most one from the <supremum norm> to $\ell_\infty$. Nonnegativity and the <subpartition of unity for B-splines> give
$$
|Bc(t)|\le\|c\|_{\ell_\infty}\sum_jN_j(t)\le\|c\|_{\ell_\infty}.
$$
Consequently $\|P_x\|\le\|A_x^{-1}\|_{\ell_\infty}$.
For the reverse bound, select a row $r$ of $A_x^{-1}$ with maximal absolute row sum, and choose $y_j=\operatorname{sgn}((A_x^{-1})_{rj})$, assigning any value of modulus at most one when the entry is zero. Then $\|y\|_{\ell_\infty}=1$ and
$$
\|A_x^{-1}y\|_{\ell_\infty}=\sum_j|(A_x^{-1})_{rj}|=\|A_x^{-1}\|_{\ell_\infty}.
$$
At distinct sites there is a continuous <piecewise linear function> $f$ with $f(x_j)=y_j$ and $\|f\|_\infty=1$: interpolate linearly between the ordered sites and extend constantly to the endpoints. The given <uniform-norm stability of a B-spline basis> therefore implies
$$
\|P_xf\|_\infty=\|BA_x^{-1}y\|_\infty\ge\frac1{d_k}\|A_x^{-1}y\|_{\ell_\infty}.
$$
Thus the <B-spline interpolation operator norm> satisfies
$$
\boxed{\frac1{d_k}\|A_x^{-1}\|_{\ell_\infty}\le\|P_x\|\le\|A_x^{-1}\|_{\ell_\infty}.}
$$
Positivity $N_i(x_i)>0$ alone, without the usual distinct-site convention, is insufficient to define an interpolation operator. For example, take order-two <B-splines> with knots $0,1,2,3$ and $x_1=x_2=3/2$. Both diagonal values are $1/2$, but the <B-spline collocation matrix> has identical rows $(1/2,1/2)$ and is singular. The displayed proof uses the well-defined interpolation map posited in the question; ordered distinct sites supply its standard existence guarantee.
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