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Possible Duplicate:
Limit of $L^p$ norm

I was asked to show:

Assume $|f|_{r}<\infty$ for some $r<\infty$. Prove that $$ |f|_{p}\rightarrow |f|_{\infty} $$ as $p\rightarrow \infty$.

I am stuck in the situation that $|f|_{p}<\infty$ for all $r

Could $f_{p}$ be fluctuating while $|f|_{\infty}=\infty$? I have proved that for $r

  • 1
    Could you write down the definition of $\vert f \vert_{\infty}$ in your post?2012-12-24
  • 0
    The usual definition that $f^{-1}(c,\infty)$ has measure 0, and $c$ is the inf of all such values possible.2012-12-24
  • 0
    I am sure this is a duplicate question, although cannot find it now.2012-12-24
  • 0
    This is from Rudin, so will not be surprising if it is a duplicate.2012-12-24
  • 2
    See [this](http://math.stackexchange.com/questions/242779/limit-of-lp-norm/242792) post.2012-12-24
  • 0
    Thanks! As I know I would not be the only one...:(2012-12-24

1 Answers 1

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I will put a partial answer to my question by combing Davide's proof and my previous work.

Suppose $|f|_{\infty}=a$, then $(r,a)\in E$ and by the continuity of $\phi$ we proved the statement. So we need to prove this under the hypothesis $|f|_{\infty}=\infty$. We have two cases:

(1) $|f|_{p}<\infty,\forall p<\infty$. Then we need to show $|f|_{p}\rightarrow \infty$ as $p\rightarrow \infty$. (2) There exist some $p>r$ such that $|f|_{p}=\infty$. We need to show $\forall q>p$, $|f|_{q}=\infty$ as well.

Now (2) is straightforward since if $|f|_{q}<\infty$, then $p\in [r,q]$ implies $|f|_{p}<\infty$ as well. This contradicts with our hypothesis. So it suffice to prove (1). But Davide already showed via a clever argument that $\lim \inf |f|_{p}\ge |f|_{\infty}$. So this force $\lim \inf |f|_{p}=\infty$ as well.