TITLE: Are the $L^p$ norms ordered by $p$?
QUESTION [5 upvotes]: A question left over from this post is:
Are the $L^p$ norms ordered by $p$ like the power means are?

REPLY [2 votes]: If $\int_\Omega  dx = 1$, then the Hölder inequality states that yes.
If $\int_\Omega  dx <\infty$, then the Hölder inequality states that there is domination between the norms. 
Otherwise, no (more details).