I am trying to see what is known about uniform density of function spaces in $C(\mathbb{R}^n)$ or $C_b(\mathbb{R}^n)$ (bounded continuous functions on $\mathbb{R}^n$). By uniform density, I mean density with respect to uniform convergence on the whole of $\mathbb{R}^n$, not any compactly supported version of uniform convergence. I do not know of any version of Stone Weierstrass theorem covering such cases. But I would be happy to see certain special examples where people have proved density of certain function spaces in $C(\mathbb{R}^n)$ or $C_b(\mathbb{R}^n)$. I am just curious about this, I do not have an especially good reason for asking this. As a possible example, if someone has proved somewhere that continuous functions of bounded smooth functions are dense in $C(\mathbb{R}^n)$, I would be interested in learning how such a proof might work. Thanks!

Comment: edited after Robert Israel and Eric Wofsey's comment.

  • 4
    $\begingroup$ "Continuous functions of real analytic functions" is $C(\mathbb R^n)$. That is, for any $f \in C(\mathbb R^n)$, we can say $f(x_1, \ldots, x_n) = f(X_1, \ldots, X_n)$ where $X_j = x_j$ is a real analytic function of $x_j$. $\endgroup$ – Robert Israel Aug 11 '15 at 17:46
  • 6
    $\begingroup$ The uniform topology on all of $C(\mathbb{R}^n)$ is highly disconnected, being the disjoint union of the cosets of $C_b(\mathbb{R}^n)$, the bounded continuous functions. So you can split the question into two parts: show that you can approximate every bounded function, and show that you can approximate every function within bounded error. The first question can (in principle) be answered using Stone-Weierstrass, since $C_b(\mathbb{R}^n)$ is the same as $C(\beta\mathbb{R}^n)$, where $\beta\mathbb{R}^n$ is the Stone-Cech compactification. $\endgroup$ – Eric Wofsey Aug 11 '15 at 17:48
  • $\begingroup$ Again, all continuous functions on $\mathbb R^n$ are continuous functions of bounded smooth functions, since $x_j = \tan(\arctan(x_j))$ and $\tan$ is continuous on $(-\pi/2, \pi/2)$. $\endgroup$ – Robert Israel Aug 11 '15 at 19:36
  • $\begingroup$ Crossposted on MSE. $\endgroup$ – Michael Albanese Aug 11 '15 at 20:58

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.