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  "body": "\n## Statement\n\nLet $0 \u003c \\lambda_1 \u003c \\lambda_2 \u003c \\cdots$ be real numbers with $\\lambda_n \\to \\infty$.\nLet $C[0, 1]$ be the space of [[continuous functions|0X3WGX95FE]] $[0, 1] \\to \\R$ with the supremum norm.\nThen the linear span of the functions $1, x^{\\lambda_1}, x^{\\lambda_2}, \\dots$ is [[dense|0BSJSX7ME2]] in $C[0, 1]$ if and only if\n$$\\sum_{n=1}^{\\infty} \\frac{1}{\\lambda_n} = \\infty.$$\n\n## Remarks\n\nThe idea of the proof: one first works in $L^2[0, 1]$.\nThere the distance from $x^m$ to the span of finitely many powers $x^{\\lambda_k}$ can be computed exactly with Gram determinants, which here are Cauchy determinants.\nThe distance is a product of factors $\\abs{m - \\lambda_k} / (m + \\lambda_k + 1)$ times a constant, and this product tends to $0$ exactly when $\\sum 1/\\lambda_k$ diverges.\nA further argument passes from $L^2$ to [[uniform approximation|XAARVNKXF1]] of the monomials $x^m$, and then the Weierstrass theorem finishes the proof.\n\nExample: the exponents $\\lambda_n = n$ give the ordinary polynomials, and $\\sum 1/n = \\infty$, which recovers the Weierstrass approximation theorem on $[0, 1]$.\nThe primes also work, since $\\sum 1/p = \\infty$, so combinations of $1$ and $x^p$ for primes $p$ are dense.\nThe squares do not: $\\sum 1/n^2 \u003c \\infty$, so the combinations of $1, x, x^4, x^9, \\dots$ are not dense in $C[0, 1]$.\n\nThe constant function $1$ is needed in the span: every $x^{\\lambda_n}$ vanishes at $0$, so without $1$ no function with $f(0) \\ne 0$ could be approximated.\n\n## Explanations\n\n### L1 human\n\nPolynomials can approximate every continuous function on $[0, 1]$.\nThe theorem asks what happens if you may use only some of the powers of $x$.\nThe answer is a single counting test: you need enough powers, in the sense that the sum of the reciprocals of the exponents is infinite.\n\n### L2 human\n\nThe theorem sharpens the Weierstrass theorem and shows how much freedom there is in choosing building blocks.\nIt is a classic application of complex analysis and Hilbert space methods to approximation; a common proof uses the zeros of bounded analytic functions in a half-plane.\nWhen the series converges, the proof shows more: no power $x^\\lambda$ with $\\lambda \u003e 0$ outside the set $\\set{\\lambda_n}$ lies in the closure of the span.\n",
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