Stone-Weierstrass Theorem

This lecture covers the Stone-Weierstrass theorem, which states that a family of functions separating and not vanishing on a compact set is uniformly dense in the space of continuous functions. The proof involves showing the convergence of polynomials to certain functions, establishing properties of separating and non-vanishing families, and demonstrating the uniform density of the generated algebra. The complex version of the theorem is also discussed, along with corollaries related to polynomial approximation in complex spaces and closed intervals in real numbers.