Stone-Weierstrass Theorem

Real Analysis · Axiom Academy

LESSON The Stone-Weierstrass Theorem When is a family of continuous functions "rich enough" to approximate every continuous function? 1. Density Means Closing the Gap A set A is dense in C(X) if every continuous f can be approached arbitrarily well by members of A . The right notion of "close" is the sup norm : the single largest vertical gap between two functions over the whole domain. the sup-norm distance — the widest gap, anywhere A is dense: every target is the limit of members of A 2. A Subalgebra: Closed Under the Operations The candidate family must be a subalgebra : take any two of its functions and the natural operations keep you inside A . It is a vector space that is also closed under pointwise multiplication. Polynomials: sums, scalar multiples, and products of polynomials are polynomials. Closure under products is what lets you build : from g(x) = x alone you can form and every polynomial. Multiplication is the source of the family's reach. 3. Separating Points (and the Constants) Two structural conditions do the heavy lifting. First, A must contain the constants . Second, A must separate points : for any there is some with . The construction: start from a separator g with , then form the combination below. Being constants + scalings + g — all the algebra operations — it lives in A , and it hits at x and at y exactly. 4. The Theorem — and Fourier Series contains the constant functions, and

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