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Vector-Valued Stone-Weierstrass Theorem?

MathOverflow Asked by mw19930312 on September 26, 2020

The standard statement of the Stone-Weierstrass theorem is:

Let $X$ be compact Hausdorff topological space, and $mathcal{A}$ a subalgebra of the continuous functions from $X$ to $mathbb{R}$ which separates points. Then $mathcal{A}$ is dense in $C(X, mathbb{R})$ in sup-norm.

Most materials that I can find on the extension of Stone-Weierstrass theorem discuss only the multivariate case, i.e., $Xin mathbb{R}^d$. I wonder whether this theorem can be extended to vector-valued continuous functions. Specifically, let $mathcal{A}$ be a subalgebra of continuous functions $Xto mathbb{R}^n$, with the multiplication defined componentwisely, i.e., $forall f, gin mathcal{A}$, $fg = (f_1g_1, ldots, f_ng_n)$. Then shall we claim $mathcal{A}$ is dense in $C(X, mathbb{R}^n)$ in sup-norm if $mathcal{A}$ separates points?

Any direct answer or reference would greatly help me!

Edit: As Nik Weaver points out, the original conjecture is false since the functions of the form $xmapsto (f(x), 0, ldots, 0)$ create a counter-example. I wonder whether there are non-trivial Weierstrass-type theorems on vector-valued functions. For instance, what if we further assume $mathcal{A}$ is dense on each `axis’?

2 Answers

This is a comment, not an answer but I am, alas, not entitled. Vector valued Stone-Weierstraß theorems were studied in great detail in the second half of the last century and there is a comprehensive monograph on the subject by João Prolla ("Weierstraß-Stone, the theorem", 1993). Not on topic, but he also considered the case of bounded, continuous vector-valued functions on non-compact spaces, using the strict topology of R.C. Buck.

Answered by bathalf15320 on September 26, 2020

I think that you want something like this:

Let $Eto X$ be a (finite rank) vector bundle over a compact, Hausdorff topological space $X$, let $mathcal{A}subset C(X,mathbb{R})$ be a subalgebra that separates points, and let $mathcal{E}subset C(X,E)$ be an $mathcal{A}$-submodule of the $C(X,mathbb{R})$-module of continuous section of $Eto X$. Suppose that, at every point $xin X$, the set ${,e(x) | einmathcal{E} }$ spans $E_x$. Then $mathcal{E}$ is dense in $C(X,E)$ with respect to the sup-norm defined by any norm on $E$.

Addendum: Here is a sketch of the argument: First, by an easy compactness argument, one can show that $mathcal{E}$ contains a finite set $e_1,ldots e_m$ such that $e_1(x),e_2(x),ldots,e_m(x)$ spans $E_x$ for all $xin X$. Then $mathcal{E}$ contains all the sections of the form $$a_1, e_1 + cdots + a_m,e_m$$ where $a_iinmathcal{A}$, and every section $ein C(X,E)$ can be written in the form $$e = f_1, e_1 + cdots + f_m,e_m$$ for some functions $f_iin C(X,mathbb{R})$. By the Stone-Weierstrass Theorem, for any given $delta>0$, we can choose $a_iin mathcal{A}$ so that $|f_i-a_i|<delta$ for all $1le ile m$. Now the equivalence of all norms in finite dimensional vector spaces can be applied (together with the compactness of $X$) to conclude that $mathcal{E}$ is dense in $C(X,E)$ in any sup-norm derived from a norm on the (finite rank) vector bundle $E$.

Answered by Robert Bryant on September 26, 2020

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