Multivariate Weierstrass theorem? The Weierstrass theorem states that for any continuous function $f$ of one variable there is a sequence of polynomials that uniformly converge to $f$. To my surprise, I couldn't find any reference to similar results (either positive or negative) for the multivariate case, i.e. when $f \in C([0, 1]^n), n > 1$.
I know about Kolmogorov's theorem but I can't see how can it apply in this case (I don't if there is a version in which the "inner" functions are just polynomials; approximating them would produce hard to quantify errors).
 A: By Stone-Weierstrass theorem (see here) aby subalgebra $A$ of $C([0,1]^n)$ that contains a constant function and separates points of $[0,1]^n$ is dense in $C([0,1]^n)$ with supremum norm ($\lVert f \rVert = \max_{x \in [0,1]^n} \lVert f(x) \rVert$ for $f$ in $C([0,1]^n)$).
Let $A$ be the set of multivariate polynomials: $A=\{\sum_{i=(i_1, \dots, i_n) \in \mathbb{N}^n} a_{i}\prod_{j=1}^n x_j^{i_j} \colon a_i \in \mathbb{R} \text{ for } i=1,2,\dots,n \}$. It is obviously a subalgebra of $C([0,1]^n)$ (all polynomials are continuous and the set $A$ is closed under addition, multiplication and multiplication by a constant). Furthermore, $A$ contains the all constant functions and separates points (if $a,b\in [0,1]^n$, then $x_i$ is a polynomial separating $a$ and $b$ where $i$ is a coordinate at which $a$ and $b$ differ). Thus $A$ is, by Stone-Weierstrass theorem, dense in $C([0,1]^n)$, so for each continuous function there is a sequence of multivariate polynomials uniformly converging to it.
