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The Stone-Weierstrass theorem

1959; American Mathematical Society; Volume: 10; Issue: 5 Linguagem: Inglês

10.1090/s0002-9939-1959-0113131-7

1088-6826

Louis de Branges,

Advanced Mathematical Modeling in Engineering

Let S be a locally compact Hausdorff space and let C(S) be the continuous complex valued functions which (in the noncompact case) vanish at infinity. Let E be a vector subspace of C(S) which is closed under complex conjugation. We ask for conditions that E be uniformly dense in C(S). (The hypothesis that E be closed under complex conjugation allows us to reduce the problem to one for real valued functions. The reader may prefer to recast the discussion in that context.) We have chosen to present our result as a lemma towards a proof of the Stone-Weierstrass theorem. This approach not only sheds an interesting light on that theorem, but will help the reader understand the nature of the lemma. However, the discussion of Stone [3] remains the most direct and, when all details are considered, the shortest proof of the Stone-Weierstrass approximation theorem. A more serious application of the lemma will be made later in a paper on the Bernstein approximation problem. Let U(E) be the set of all real valued measures A on the Borel subsets of S, with total variation at most 1, such that for every f in E, ffdlu = 0. Consider U(E) in the weak topology induced by C(S) under integration. Then U(E) is a compact, convex set, and if E is not dense in C(S), U(E) contains a nonzero element (Loomis [2, pp. 22-23, pp. 29-47]). By the Krein-Milman theorem, U(E) is the weakly closed convex span of its extreme points (Krein-Milman [11).