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| | 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. |

| | Further, there is a generalization of the Stone-Weierstrass theorem to noncompact Tychonoff **spaces**, namely, any continuous function on a Tychonoff **space** **space** is approximated uniformly on **compact** sets by algebras of the type appearing in the Stone-Weierstrass theorem and described below. |

| | Suppose K is a **compact** **Hausdorff** **space** with at least two points and L is a lattice in C(K,R). |

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