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  "body": "\n## Statement\n\nLet $K$ be a [[compact|0TVJZYB6SH]] [[Hausdorff space|0S1FJ8VEX9]] with at least $n$ points.\nLet $C(K)$ be the space of continuous functions $K \\to \\R$ with the supremum norm.\nLet $V \\subseteq C(K)$ be a linear subspace of dimension $n \\ge 1$.\nThen the following are equivalent.\n\n1. Every $f \\in C(K)$ has exactly one [[best approximation|7N4988HN3P]] from $V$ in the supremum norm.\n2. $V$ is a Haar subspace: every basis of $V$ is a [[Chebyshev system|NY5AK617TR]], that is, every nonzero $v \\in V$ has at most $n - 1$ zeros in $K$.\n\n## Remarks\n\nThe idea of the proof: best approximations always exist, since $V$ is finite-dimensional.\nIf $V$ is Haar and $v_1, v_2$ are both best, then so is $(v_1 + v_2)/2$.\nIts error attains the norm only at points where $v_1 - v_2$ vanishes.\nFor a Haar subspace the error of a best approximation attains its norm at $n + 1$ or more points, too many zeros for $v_1 - v_2 \\ne 0$.\nConversely, if a nonzero $v \\in V$ vanishes at $n$ points, one builds a continuous $f$ that has $0$ and all small multiples of a suitable element of $V$ as best approximations.\n\nExample: the polynomials of degree at most $n - 1$ on $[a, b]$ form a Haar subspace, so best [[uniform approximation|XAARVNKXF1]] of a continuous function by polynomials of a fixed degree is unique.\n\nNon-example: on $[-1, 1]$, the span of $1$ and $x^2$ is not Haar, since $x^2 - 1$ has two zeros.\nCorrespondingly $f(x) = x$ has many best approximations from it.\nFor every $g = a + c x^2$, the error $x - g$ takes the values $1 - a - c$ and $-1 - a - c$ at $\\pm 1$, which differ by $2$, so $\\norm{x - g}_\\infty \\ge 1$.\nFor $\\abs{c} \\le 1/2$, the error $h(x) = x - c(x^2 - 1)$ has $h' = 1 - 2cx \\ge 0$ on $[-1, 1]$ and $h(\\pm 1) = \\pm 1$, so $\\norm{h}_\\infty = 1$.\nHence every $c(x^2 - 1)$ with $\\abs{c} \\le 1/2$ is a best approximation of $x$.\n\n## Explanations\n\n### L1 human\n\nIn the uniform norm, best approximations are not always unique.\nHaar's theorem says exactly when they are: when no nonzero function of the approximating space has too many zeros.\nPolynomials of a fixed degree pass this test, which is why the best uniform polynomial approximation is unique.\n\n### L2 human\n\nThe theorem explains the special role of polynomials and of Chebyshev systems in uniform approximation.\nFor Haar subspaces the best approximation is characterized by alternation of the error at $n + 1$ points, which generalizes the equioscillation theorem.\nThe Mairhuber–Curtis theorem shows that Haar subspaces of dimension at least $2$ do not exist on open sets in several variables.\nSo best uniform approximation by multivariate polynomials of a fixed degree is not always unique.\n",
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