How (and when) to factor a function defined on a product of metric spaces? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T10:06:29Z http://mathoverflow.net/feeds/question/101147 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/101147/how-and-when-to-factor-a-function-defined-on-a-product-of-metric-spaces How (and when) to factor a function defined on a product of metric spaces? Niccolo' 2012-07-02T14:21:55Z 2012-07-02T23:36:49Z <p>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.</p> <p>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.</p> http://mathoverflow.net/questions/101147/how-and-when-to-factor-a-function-defined-on-a-product-of-metric-spaces/101188#101188 Answer by Nik Weaver for How (and when) to factor a function defined on a product of metric spaces? Nik Weaver 2012-07-02T23:36:49Z 2012-07-02T23:36:49Z <p>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 ${\bf Q}_p^n$.</p>