I'm reading a book where it says that if $q>p$ then $H^q([0,2\pi])$ is dense in $H^p([0,2\pi])$, and that this is due to the fact that the trigonmetric polynomials are dense in $H^p([0,2\pi])$.
I know how to show that the trigonmetric polynomials are dense in $H^p([0,2\pi])$, but I don't know how to formally use this to get that $H^q([0,2\pi])$ is dense in $H^p([0,2\pi])$. If 'feels' like this is a true statement, but I really would like to show it?
Let $P$ be the set of trigonometric polynomials. Clearly $P$ is a subset of each Sobolev space. You've got a proof that $P$ is dense in $H^p([0,2\pi])$, so can you prove the second inclusion of $$P \subset H^q([0,2\pi]) \subset H^p([0,2\pi])?$$ Since $P$ is dense so is the middle space.