I'm reviewing Rudin's Principles of analysis now.
He first proves Weierstrass approximation and uses it to generalize to Stone-Weierstrass theorem.
The problem is, this version is very restrictive.
What he proves here is that:
Let $X$ be a compact metric space.
Let $\mathscr{A}$ be a self-adjoint algebra of $C(X,\mathbb{C})$.
Let $C(X,\mathbb{C})$ be equipped with the sup norm.
If $\mathscr{A}$ separates points in $X$ and vanishes at no point of $X$, then $\mathscr{A}$ is dense in $C(X,\mathbb{C})$.
However, the general version in wikipedia states that this is true when $X$ is just compact Hausdorff.
I tried to remove metric hypotheses in Rudin's argument, but since his argument is based on second-countability of a compact metric space, I couldn't.
In general, compact Hausdorff is NOT second countable. So I guess that Rudin's argument cannot generalize it further.
Am I thinking right?
If I'm thinkihg right, then i have the following question.
Textsbooks in my hand right now are Munkres-Topology,Rudin-PMA,RCA, but there is no chapter for this general version of Stone-Weierstrass Theorem in those texts.
So, is there any text introducing this theorem in full strength? And is the proof quite elementary so that an undergraduate (just like me) can understand the proof?
Best Answer
Sticking with Rudin, try his "Functional Analysis", section 5.7, for a very slick proof (due to Glicksberg).