= Extension of approximation operators from a dense subspace
{title2=$\|T_nf-f\|\le(C+1)\|f-p\|+\|T_np-p\|$}
= Lebesgue convergence criterion for approximation operators
{c}
{synonym}
Suppose <linear operators> $T_n$ on a <normed vector space> satisfy $\|T_n\|\le C$ uniformly and $T_np\to p$ on a <dense subspace>. The displayed <triangle inequality> for any element $p$ of that subspace first permits $p$ to approximate $f$, then lets $n$ increase, proving $T_nf\to f$ for every $f$. Applied to the <Bernstein polynomial>, the <Weierstrass approximation theorem> supplies the dense polynomial subspace and $C=1$.
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