Lipschitz functions dense in C0M

curtdbz

I'm working on Pugh's book on analysis and there's this problem that should be very easy to solve. It's asking to show that the set of continuous functions, [tex]f:M \rightarrow R, f\in C^{Lip}[/tex] obeying the Lipschitz condition (where M is a compact metric space):

[tex]|f(a) - f(b)| \leq L d(a,b)[/tex] for some L, for every a and b belonging to to M.

Show that the above is dense in [tex]C^{0}(M,R)[/tex]. My attempt is to use the Stone-Weirestrass theorem. That is to show that the set [tex]C^{Lip}[/tex] vanishes nowhere and separates points. The latter is easy for me, I just showed how the above equation implies that f(a) does not equal f(b) if a does not equal b. However, showing the vanishing property is proving difficult. Is there some trick I'm supposed to use? Hm...

Also, I assume I'll have to actually show that [tex]C^{Lip}[/tex] is infact a function algebra; that is, it obeys the 3 properties that makes something that (closed under addition, constant multiples, and multiplication), but I can't seem to manipulate the equations in such a way that shows the Lipschitz property implies those.

Any help is appreciated. Thank you.

JSuarez

The Stone-Weierstrass theorem is applicable to subalgebras, so you must show first that the LIpschitz functions are indeed a subalgebra of [itex]C^{0}\left(M,\mathbb R \right)[/itex]. This is not difficult, for the multiplication, start with:

[tex]\left|f\left(a\right)g\left(a\right)-f\left(b\right)g\left(b\right)\right|[/tex]

Then add and subtract [itex]f\left(a\right)g\left(b\right)[/itex], expand and use the fact that the functions are continuous and defined in a compact space.

Now, there are some things I don't understand in your question:

This is indeed necessary for the locally compact version of the theorem but, as you have a compact metric space, you only need to show that your set contains the constant functions.

How did you prove that?