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  "body": "\n## Statement\n\nLet $X$ be a [[normed vector space|03J5RAKB30]] and let $V \\subseteq X$ be a linear subspace.\nLet $P \\colon X \\to X$ be a bounded linear map with image in $V$ and $P v = v$ for all $v \\in V$, with [[operator norm|06EKW1AKXF]] $\\norm{P}$.\nThen for every $f \\in X$,\n$$\\norm{f - P f} \\le (1 + \\norm{P}) \\inf_{v \\in V} \\norm{f - v}.$$\n\n## Remarks\n\nProof: for any $v \\in V$, $P v = v$ gives $f - P f = (f - v) - P(f - v)$.\nSo $\\norm{f - P f} \\le (1 + \\norm{P}) \\norm{f - v}$, and taking the infimum over $v$ gives the claim.\n\nExample: let $P$ be polynomial interpolation at $n + 1$ nodes in $[a, b]$, acting on $C[a, b]$ with the uniform norm.\nThen $\\norm{P}$ is the [[Lebesgue constant|7V7KWNK9HN]] $\\Lambda_n$, and the lemma gives $\\norm{f - P f}_\\infty \\le (1 + \\Lambda_n) E_n(f)$, where $E_n(f)$ is the error of best uniform approximation by polynomials of degree at most $n$.\nFor Chebyshev nodes, $\\Lambda_n \\le \\frac{2}{\\pi} \\log(n + 1) + 1 \u003c 4$ for $n \u003c 100$, so interpolation loses less than a factor $5$ compared with the best polynomial.\n\nThe same lemma applies to Fourier partial sums, with $\\Lambda_n$ the Lebesgue constants of the Dirichlet kernel.\n\n## Explanations\n\n### L1 human\n\nA linear [[projection|033DA1WE8Y]] onto a space of approximants is never much worse than the best approximation from that space.\nThe loss factor is at most one plus the norm of the projection.\nSo a cheap method like interpolation is nearly optimal whenever its Lebesgue constant is small.\n\n### L2 human\n\nThis is how Lebesgue constants enter convergence proofs.\nCombined with Jackson's inequality, it shows that interpolation at Chebyshev nodes converges uniformly when $\\omega(f, 1/n) \\log n \\to 0$, the Dini–Lipschitz condition.\nConversely, unbounded norms of projections lead through the uniform boundedness principle to divergence results such as Faber's theorem.\n",
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