Stone-Weierstrass Theorem in $\mathbb{C}$

Question

I am having difficulty understanding how to prove the Stone-Weierstrass Theorem for complex valued functions defined on the closed unit disc $\mathbb{D}\subset\mathbb{C}$.

Here is a version I have from an exercise in Lang:

Any continuous complex valued function defined on the closed unit disc can be uniformly approximated by polynomials.

I take this to mean that for any continuous $f:\mathbb{D}\to\mathbb{C}$ there is a sequence of polynomials $\{f_n\}_{n\in\mathbb{Z}^+}$ such that for any $\epsilon>0$, there is $N\in\mathbb{Z}^+$ so that for all $n\geq N, \sup_{z\in\mathbb{D}}|f_n(z)-f(z)|<\epsilon$.

Is this a valid interpretation? What would be the best way to approach this? Ideally I would like to use tools from elementary complex analysis but any insights could be helpful! :)

Answer 1

Is this a valid interpretation?

No, because the set of polynomials in z is not self conjugate. If you have a series of polynomials in z that converges in the supremum norm on D, the limit function needs to be holomorphic again, showing that an arbitrary continuous function cannot be approximated by such polynomials: All continuous functions would need to be holomorphic.

In order to apply the Stone-Weierstrass theorem, you'd need to consider polynomials in z and $\bar z$. Let $h(z)$ be a continuous function. Then we can write $$ h(z) = f(z) + i g(z) $$ with real valued functions f and g. These can be approximated as real valued functions with polynomials $p_f (x, y)$ and $p_g(x, y)$ in $x, y$ by the real version of the Stone-Weierstrass theorem. Any polynomial in $x, y$ can be transformed into a polynomial in the variables z and $\bar z$, so that we can make $$ \| h(z) - (p_f(z, \bar z) + i p_g(z, \bar z)) \|_{\sup} $$ arbitrarily small.

(This is the proof of the complex version using the real version of the Stone-Weierstrass theorem given by Lang applied to this concrete situation.)

Since the Stone-Weierstrass theorem is essentially about continuous functions and not about holomorphic ones, I doubt that there are any neat tricks from complex calculus that could make the general proof given by Lang easier, shorter, more elegant in this particular case.

Note: We are referring to the book

• Serge Lang: "Real and Functional Analysis"