Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 339 / 3 / b / Solution

Let $E=\mathbb{R}[x{]}_{\le d}$ with the supremum norm. By equivalence of norms in finite dimensions, there is $C_d$ such that ${\sum }_{j=0}^d|a_j|\le C_d\Vert p{\Vert }_{\infty }$ whenever $p={\sum }_j{a}_j{x}^j$. Henceso ${\epsilon }_n:=\Vert B_n-I{\Vert }_{\mathrm{op}}\le C_d{\max }_{0\le j\le d}\Vert B_n(x^j)-x^j{\Vert }_{\infty }\to 0$. This upgrades convergence on fixed polynomials to operator norm convergence on the fixed space.

For sufficiently large $n$, ${\epsilon }_n<1$, and a Neumann series givesThus the inverse Bernstein approximation on a fixed-degree polynomial space satisfies $\Vert g_n-f{\Vert }_{\infty }\to 0$. By the extreme value theorem, strict positivity on the compact interval gives $\mu ={\min }_{[0,1]}f>0$. Eventually $\Vert g_n-f{\Vert }_{\infty }<\mu /2$, and consequentlyApplying pointwise convergence to the varying polynomials $g_n$ without this finite-dimensional operator argument would not justify the conclusion.