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Write $Ba={\sum }_{j=1}^n{a}_j{N}_j$, and let $Rf=(f(x_1),\dots ,f(x_n){)}^T$. 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 henceNonnegativity and the subpartition of unity for B-splines giveFor 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 $\Vert Rf{\Vert }_{{\ell }^{\infty }}\le \Vert f{\Vert }_{\infty }$, the operator norm upper bound isHere 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 $\Vert y{\Vert }_{{\ell }^{\infty }}=1$ andA 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:ThereforeThe construction of the bounded continuous data extension is what permits the matrix norm to give a lower bound for a function-space operator norm.