1

I'm trying to prove that $C_c^{k}(\mathbb{R}^n)=C_c(\mathbb{R}^n) \cap C^{k}(\mathbb{R}^n)$ is dense in $L^p(\mathbb{R}^n) \cap C^{k}(\mathbb{R}^n)$.

I know that $C_c(\mathbb{R}^n)$ is dense in $L^p(\mathbb{R}^n)$ so $C_c(\mathbb{R}^n)$ is dense in $L^p(\mathbb{R}^n) \cap C^{k}(\mathbb{R}^n)$.

$C^{k}(\mathbb{R}^n)$ is trivially dense in $L^p(\mathbb{R}^n) \cap C^{k}(\mathbb{R}^n)$.

Using that result Intersection of Dense Sets, I see that it is sufficient that one of the dense sets is open. How can I prove it?

Pao
  • 59
  • What norm on the intersection are you using? sum of norms? – Calvin Khor Mar 26 '21 at 11:23
  • Can't I use the $| \cdot |_p$ norm on the intersection? – Pao Mar 26 '21 at 11:33
  • This should follow from the density of $C_c^{\infty}(\mathbb{R}^n)$ in $L^p(\mathbb{R}^n)$, no? Also the statement "$C^k(\mathbb{R}^n)$ is trivially dense in $L^p(\mathbb{R}^n)\cap C^k(\mathbb{R}^n)$" is incorrect. – Cameron Williams Mar 26 '21 at 11:35
  • You can, but then its not a Banach space, as its completion is of course $L^p$. The sum of norms makes the intersection of Banach spaces into another Banach space – Calvin Khor Mar 26 '21 at 11:37
  • @cameron-williams why? – Pao Mar 26 '21 at 11:43
  • Are polynomials in $L^p(\mathbb{R}^n)$? – Cameron Williams Mar 26 '21 at 11:44
  • @CameronWilliams no, $f(x)=x^2$ for example is not in $L^2(\mathbb{R})$ because $\int_{\mathbb{R}} \left(\lvert x^{2}\rvert \right)^{2} , dx = \infty$ – Pao Mar 26 '21 at 11:53
  • Correct. And polynomials are in $C^k$, so you can see that your statement is false. – Cameron Williams Mar 26 '21 at 11:54
  • Take $f \in L^p(\mathbb{R}^n) \cap C^{k}(\mathbb{R}^n)$ then $f_n=(1-\dfrac{1}{n}) f$ is in $ C^{k}(\mathbb{R}^n)$ and $f_n \rightarrow f$ for $n \rightarrow \infty$. Now $\lvert f_n - f \lvert^p= \lvert f_n\lvert ^p \leq \lvert f \lvert ^p \in L^1(\mathbb{R}^n)$. Now I can use Dominated Convergence Theorem and then $\lim_{n \rightarrow \infty} | f_n-f |_p^p=0$. – Pao Mar 26 '21 at 12:15
  • No, neither of those sets is open. – David C. Ullrich Mar 26 '21 at 13:02
  • @CalvinKhor What norm on $C^k(\mathbb{R}^n)$ are you referring to? These functions are not bounded, need not be integrable.. – Andrew McMillan Mar 27 '21 at 16:15
  • I assumed the notation of eg Evans so sum of sup norms of derivatives which makes the set different from the vector space of all $C^k$ functions but you’re right maybe OP did not mean it @AndrewMcMillan – Calvin Khor Mar 28 '21 at 02:14

0 Answers0