Inverse Bernstein approximation on a fixed-degree polynomial space ($\Vert B_n^{-1}f-f{\Vert }_{\infty }\to 0$)

On the fixed finite-dimensional space of polynomials of degree at most $d$, uniform convergence of $B_n$ on each monomial and equivalence of norms in finite dimensions imply ${\epsilon }_n=\Vert B_n-I{\Vert }_{\mathrm{op}}\to 0$. A Neumann series then gives $\Vert B_n^{-1}-I{\Vert }_{\mathrm{op}}\le {\epsilon }_n/(1-{\epsilon }_n)$ for sufficiently large $n$. Consequently the displayed convergence holds. A strictly positive polynomial has positive minimum on the compact interval, so its inverse Bernstein approximants are also strictly positive for all sufficiently large $n$.