I'm trying to summarize what I know about $L^p$ spaces at the moment and have a few questions. I think that for $1\le p\le q\le \infty$ one has $$\mathscr{l}^p(\mathbb{N})\subseteq\mathscr{l}^q(\mathbb{N}) \text{ and for all }x\in\mathscr{l}^p(\mathbb{N}): \| x\|_q\le\| x \|_p$$

while for a finite measure space $(X,\mu)$ $$L^q\subseteq L^p \text{ and for all }f\in L^q:\| f\|_p\le\mu(X)^{\frac{1}{p}-\frac{1}{q}}\| f\|_q$$

My questions: 1. Are the above statements correct?

  1. Are the two statements about sequence spaces also true if I consider $\mathbb{Z}$ instead of $\mathbb{N}$?

  2. Are the space-inclusions 'proper' for $p<q$?

  3. For the non-sequential $L^p$ spaces: What if my measure isn't finite. Is there still something I can say about embeddings and norm-inequalities? I tried Wikipedia but couldn't make much sense of what I read.

  4. Is there anything else important I should know about $L^p$ spaces? I'm talking very basic stuff that may be useful for an undergraduate writing a functional analysis exam. Save inequalities, I know most of these already.

I know that's quite a lot but I tried to find answers to these questions for quite a while without much success.

  • 1
    $\begingroup$ $\ell^p(\Bbb Z)\cong\ell^p(\Bbb N)$, just look at a bijection between the indexing sets. Absolute convergence means this is valid. $\endgroup$ – Adam Hughes Sep 25 '15 at 17:54
  • $\begingroup$ Note: you can use \| instead of \lvert, \rvert, etc. for proper spacing $\endgroup$ – Math1000 Sep 25 '15 at 18:25
  • $\begingroup$ @AdamHughes Yeah I see it now, thank you. $\endgroup$ – azureai Sep 26 '15 at 11:47
  • $\begingroup$ @Math1000 Where do you think that the spacing could be improved, for example? $\endgroup$ – azureai Sep 26 '15 at 11:47
  • 1
    $\begingroup$ Compare e.g. $||x||$ to $\|x\|$. $\endgroup$ – Math1000 Sep 26 '15 at 22:52

For your 4th question:

Suppose $f(x)=x^\alpha, x\in (1,\infty)$

It is easy to show that, $$f\in L^p(1,\infty)\Leftrightarrow \alpha< -\frac{1}{p}$$

Here $\mu(X)=\infty$. Take $p=1$ and $q=2$ and take $\alpha$ such that $-1<\alpha <-\frac{1}{2},$ then

$\ f \in L^2(\mu) $ but $f \notin L^1(\mu)$. So above result will not be true if measure of the space is not finite.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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