Stone-weierstrass theorem

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This chapter describes Stone-Weierstrass theorem. A subset A ⊂ C(X; F) is said to be separating over X, if A satisfies property x # y in x, when there is some element a ∊ A, such that a(x) = 1 and a(y) = 0. It follows that, when (F,|•|) is non-archimedean, and A is a separating subset of C(X;F), then the unitary subalgebra generated by A satisfies property. A subset A ⊂ C(X; F) is said to be duel if its K-closure contains the set of all F–characteristic functions of subsets of X, which are both open and closed. If (E,τ) is a T V S over non-archimedean valued division ring (F,|•|) such that E′ is separating over E. Then Pf(E) ⊗ E is k-dense in C(E;F). © 1982, North-Holland Publishing Company

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