Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Is the Sobolev space $H^s(\mathbb{R}^3)$ dense in $L^2(\mathbb{R}^3)$ for $s>0$ ?

share|cite|improve this question

We denote by $\mathcal D(\Bbb R^d)$ the set of smooth functions with compact support.

  • As the Fourier transform of a function in $\mathcal D(\Bbb R^d)$ has an arbitrary decay, we have the inclusion $\mathcal D(\Bbb R^d)\subset H^s(\Bbb R^d)$ for any $s>0$.
  • By Plancherel's formula, we have $H^s(\Bbb R^d)\subset L^2(\Bbb R^d)$.
  • By a truncation and regularization argument, it can be shown that $\mathcal D(\Bbb R^d)$ in dense in $L^2(\Bbb R^d)$ with its natural norm.
share|cite|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.