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Let $1\leq p \leq q \leq \infty$.

It's well known that on a finite measure space $(X,\mathcal{M}, \mu)$, we have the inclusion $L^q(X,\mathcal{M}, \mu) \subset L^p(X,\mathcal{M}, \mu)$. Questions regarding this have already been addressed ( $L^p$ and $L^q$ space inclusion). But we have also have special cases, for instance if $\mu$ is the counting measure then $L^p(X,\mathcal{M}, \mu) \subset L^q(X,\mathcal{M}, \mu)$. On this site I also found this: When $L^p \subset L^q$ for $p <q$..

So here's my question: what are some other cases of $L^p$ space inclusion? Also, what is the nature of this inclusion? Are there examples/criteria where $L^p$ is, say, open, closed (complete), dense, compact, etc. in $L^q$ for $p\neq q$. Of course, here we're viewing $L^p$ functions with the $L^q$ norm.

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GiantTortoise1729
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1 Answers1

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If $X$ is a subspace of a (normed) vector space $Y$, then

  1. $X$ is convex.
  2. $X$ is open in $Y$ iff $X=Y$.
  3. $X$ is compact iff it is bounded iff $X$={0}$.

This answers some of your questions.

Now, $L^p$ always contains the simple functions which vanish outside a set of finite measure. For $q\neq \infty$ (or if the measure space is finite), these are dense in $L^q$. Hence, if $L^p \subset L^q$, this inclusion is dense under the assumptions just stated.

Thus, under the assumptions from above ($q<\infty$ or finite measure space and $L^p \subset L^q$), $L^p$ is closed in $L^q$ if and only if $L^p = L^q$, which almost never holds (unless the set $X$ is finite or something similar holds).

PhoemueX
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