I'm taking a course of mathematical analysis and my professor just told us that Stone-Weierstrass theorem it's really important, but he didn't say why.
Can someone explain the idea of the proof and how to visualize the theorem.
And why is so important?
Let A $\subset C(K)$ such that
1) A is a subalgebra with unity 1
2) For each $\ x_1, x_2 \in K $ with $\ x_1 \neq x_2 $, exist $f \in A$ such that f($\ x_1$) $ \neq $ f($\ x_2$).
Then $ \overline A = C(K)$, where C(K) is the space of continuous functions over a compact space