Coefficient condition number of a normalized B-spline basis ($\kappa (\mathcal{S})={\sup }_{s\ne 0}\Vert c(s){\Vert }_{{\ell }^{\infty }}/\Vert s{\Vert }_{\infty }$)

For a nonnegative partition-normalized B-spline basis, the synthesis linear map $Tc={\sum }_j{c}_j{N}_j$ has operator norm one. Its condition number is consequently $\Vert T^{-1}\Vert$, with the supremum norm on the spline space and the maximum norm on its coefficients. The uniform-norm stability of a B-spline basis bounds this number independently of the spline knot sequence when the order is fixed. This is different from the condition number of a particular sampled B-spline collocation matrix.