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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.

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    $\begingroup$ "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
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    $\begingroup$ 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

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