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  "body": "\n## Definition\n\nLet $K$ be a [[compact|0TVJZYB6SH]] [[Hausdorff space|0S1FJ8VEX9]] with at least $n$ points, and let $u_1, \\dots, u_n \\colon K \\to \\R$ be [[continuous functions|0X3WGX95FE]].\nThe family $(u_1, \\dots, u_n)$ is a *Chebyshev system* (or *Haar system*) on $K$ if the following holds.\nEvery linear combination $c_1 u_1 + \\cdots + c_n u_n$ with $(c_1, \\dots, c_n) \\ne 0$ has at most $n - 1$ zeros in $K$.\nAn $n$-dimensional subspace of $C(K)$ is a *Haar subspace* if it has a basis that is a Chebyshev system.\n\n## Remarks\n\nEquivalently, the [[determinant|00KDZRW4C0]] $\\det \\big( u_i(x_j) \\big)_{i, j = 1}^{n}$ is nonzero for all distinct $x_1, \\dots, x_n \\in K$.\nEquivalently, for all distinct $x_1, \\dots, x_n \\in K$ and all $y_1, \\dots, y_n \\in \\R$ there is exactly one combination $u = \\sum c_i u_i$ with $u(x_j) = y_j$ for all $j$.\n\nExample: $1, x, \\dots, x^{n-1}$ on $[a, b]$.\nA nonzero polynomial of degree at most $n - 1$ has at most $n - 1$ roots.\n\nNon-example: $1, x^2$ on $[-1, 1]$, with $n = 2$.\nThe combination $x^2 - 1$ is nonzero but has two zeros, $\\pm 1$, more than $n - 1 = 1$.\n\nThe trigonometric polynomials of degree at most $m$ form a Haar subspace of dimension $2m + 1$ on the circle $\\R / 2\\pi\\Z$: a nonzero one has at most $2m$ zeros in a period.\n",
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