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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).

matb
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    That's strange that you couldn't find a reference. Every continuous function on a compact subset of any $\mathbb{R}^n$ can be uniformly approximated by polynomials. You may have more luck if you look for the Stone-Weierstraß theorem (which is even more general). – Daniel Fischer Nov 24 '14 at 10:04
  • @DanielFischer Thanks, that seems to be it. So general it takes a while to see it actually covers my case. – matb Nov 24 '14 at 10:18
  • Dear matb, I am now facing the same question, a reference for the multivariate case of Weierstras theorem. Did you found any reference, can you propose some for me? thank you in advance. – Nizar Sep 09 '17 at 08:47
  • @Nizar I've just answered my question. I can add more to the answer if it is not detailed enough for you. – matb Sep 10 '17 at 11:11
  • @Nizar and future readers, I think that "A course in approximation theory" by Cheney and Light from 2009 (AMS) in page 151 offer a proof of that theorem – user1868607 Apr 08 '18 at 14:53

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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.

matb
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