1
$\begingroup$

Let $ \mathscr{D}=\mathscr{D}(\mathbb{R}^n - {0}) $ be the space of smooth functions with compact support in $ \mathbb{R}^n - {0} $ topologized by the standard Schwartz topology and let $ \mathscr{C} \subset \mathscr{D} $ be the linear subspace generated by the functions of the type:

$ f(x) = \theta(|x|) \psi(x)\;\; \; x \in \mathbb{R}^n - {0} $

where $ \psi(x) = \psi(rx) $ whenever $ r >0 $ and $ \theta $ is a smooth function with compact support on the real line such that $ 0 $ does not belong to the support of $ \theta$.

Then I would like to prove that this space is dense in $ \mathscr{D} $.

Note: it is a consequence of Stone Weierstrass theorem that $ \mathscr{C} $ is dense in the space of all continuous functions with compact support in $ \mathbb{R}^n - {0} $ with respect to the topology of uniform convergence.

$\endgroup$
  • $\begingroup$ Maybe you wanted to write $\mathbb{R}^n - \{0\}$ instead of $\mathbb{R}^n - {0}$? Markup: $\mathbb{R}^n - \{0\}$ vs. $\mathbb{R}^n - {0}$. $\endgroup$ – Martin Sleziak Jul 28 '14 at 12:03
1
$\begingroup$

Using the mapping $\mathbb R\times S^{n-1}\ni(r,\omega)\mapsto e^r\omega\in\mathbb R^n\setminus\{0\}$, you can identify $\mathbb R^n\setminus\{0\}=\mathbb R\times S^{n-1}$. You have two smooth manifolds $M$ and $N$ and you can ask if $\mathscr D(M)\otimes\mathscr D(N)$ is dense in $\mathscr D(M\times N)$. Here $\mathscr D(M)\otimes\mathscr D(N)\subset\mathscr D(M\times N)$ is spanned by the products $(u\otimes v)(x,y)=u(x)v(y)$. This is generally true; see this question for reasons.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy