Suppose we have a set of regular functions defined on a product of metric spaces, for instance the Banach space of the smooth functions from $\mathbb R^n$ to $\mathbb C$. We know, thanks to the Taylor series, that such a function can 'almost' be written as a linear combination of products $f\_1 \cdot \ldots \cdot f\_n$, each of the $f\_i$ a continuous function from $\mathbb R$ to $\mathbb C$, where the almost means we are using a density argument in the sup norm.


Is there a way to generalize such a situation to a generic product of metric spaces? In particular I am interested in the space of locally constant, compactly supported functions on powers of the $p$-adic field $\mathbb Q\_p$ but also the general question sounds interesting. Any reference is greatly appreciated, thanks.



How (and when) to factor a function defined on a product of metric spaces?
Is this the statement you want: any locally constant, compactly supported function from ${\bf Q}\_p^n$ to ${\bf C}$ is uniformly approximated by linear combinations of products $f\_1\cdots f\_n$ where each $f\_i$ is a locally constant, compactly supported function from ${\bf Q}\_p$ to ${\bf C}$? Yes, this follows from the Stone-Weierstrass theorem for locally compact Hausdorff spaces. The linear combinations in question constitute a self-adjoint algebra of functions which separate points and separate each point from infinity. Therefore they are uniformly dense in the space of continuous functions vanishing at infinity on ${\bf Q}\_p^n$.


