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Stone–Weierstrass theorem - Wikipedia, the free encyclopedia (951 words) |
 | In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on an interval [a,b] can be uniformly approximated as closely as desired by a polynomial function. |
| The Stone–Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval [a,b], an arbitrary compact Hausdorff space K is considered, and instead of the algebra of polynomial functions, approximation with elements from more general subalgebras of C(K) is investigated. |
| As a consequence of the Weierstrass approximation theorem, one can show that the space C[a,b] is separable: the polynomial functions are dense, and each polynomial function can be uniformly approximated by one with rational coefficients; there are only countably many polynomials with rational coefficients. |
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