Polynomial reproduction error bound ($\Vert f-Pf\Vert \le (1+\Vert P\Vert )E_V(f)$)

If a bounded linear map $P$ reproduces every element of an approximating polynomial space $V$, then $f-Pf=(f-v)-P(f-v)$ for every $v\in V$. The triangle inequality and infimum over $v$ prove the displayed error bound. The same argument works for any linear approximating space, and applies in particular to de la Vallée Poussin sums.