I am studying Stone-Weierstrass Theorem. I wonder whether A is equal to the set of polynomials? If so, how can I proof this? And the statement is as follows:
Let $S$ be a compact set, and let $A$ be an algebra of real valued functions on S. Assume that $A$ separates points and contains the constant functions. Then, the uniform closure of $A$ is equal to the algebra of all real continuous functions on $S$.
The point is that the theorem works for any such algebra. That's why it extends the original result of Weierstrass.
If in particular $S\subset \mathbb R$, the theorem applies if you take $A$ to be the polynomials (because they separate points and contain the constant functions).