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  "body": "\n## Statement\n\nLet $d \\ge 2$ and let $\\Omega \\subseteq \\R^d$ have an [[interior point|0RZH6W98HK]].\nLet $n \\ge 2$ and let $u_1, \\dots, u_n \\colon \\Omega \\to \\R$ be continuous.\nThen some nonzero linear combination of $u_1, \\dots, u_n$ has at least $n$ zeros in $\\Omega$.\nSo $(u_1, \\dots, u_n)$ is not a [[Chebyshev system|NY5AK617TR]] on any set containing these zeros, and $\\Omega$ carries no Haar subspace of dimension $n \\ge 2$.\n\n## Remarks\n\nThe idea of the proof: suppose $(u_1, \\dots, u_n)$ is a Chebyshev system, so $D(x_1, \\dots, x_n) = \\det \\big( u_i(x_j) \\big)$ is nonzero for distinct points.\nChoose distinct $x_1, \\dots, x_n$ in a small open ball inside $\\Omega$.\nBecause $d \\ge 2$, one can move $x_1$ and $x_2$ continuously inside the ball so that they exchange places, without colliding with each other or with the other points.\nAlong this motion $D$ changes continuously from $D(x_1, x_2, \\dots)$ to $D(x_2, x_1, \\dots) = -D(x_1, x_2, \\dots)$, so by the intermediate value theorem it vanishes somewhere, a contradiction.\n\nExample: the polynomials of degree at most $1$ in two variables are spanned by $1, x, y$.\nThey are not a Haar space on the unit square: the nonzero function $x - y$ vanishes on the whole diagonal.\n\nIn dimension $d = 1$ the argument fails, because two points of an interval cannot exchange places without meeting.\nIndeed the polynomials of degree at most $n - 1$ form a Haar subspace there.\n\n## Explanations\n\n### L1 human\n\nOn an interval you can always fit a unique polynomial of degree $n - 1$ through $n$ points, whatever the points are.\nIn the plane or in space nothing of this kind is possible for a fixed family of functions.\nFor some placement of the points the interpolation problem breaks down.\nThe reason is that two points in the plane can swap places by walking around each other.\n\n### L2 human\n\nThe theorem is why multivariate interpolation and uniform approximation behave so differently from the one-variable theory.\nUniqueness of best uniform approximation and unisolvence for all point sets cannot hold for a fixed space of functions when $d \\ge 2$.\nPractical methods respond by letting the space depend on the data points, as in [[radial basis function interpolation|XZ33P3HBB9]], where the basis functions are centred at the data points.\nThe theorem holds more generally for compact Hausdorff spaces: a Haar subspace of dimension at least $2$ exists only if the space embeds in a circle.\n",
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