# Ask for two counterexamples regarding convergence in L^p

• Jun 10th 2010, 03:30 AM
zzzhhh
Ask for two counterexamples regarding convergence in L^p
If $\displaystyle <f_n>$ is a sequence in $\displaystyle L^p$ converging in $\displaystyle L^p$ to an f in $\displaystyle L^p$, that is, $\displaystyle \|f_n-f\|_p\to 0$ as $\displaystyle n\to\infty$, do we necessarily have $\displaystyle \lim\limits_{n\to\infty}\|f_n\|_p=\|f\|_p$? If not, could you please come up with a counterexample?
If $\displaystyle <f_n>$ is a sequence in $\displaystyle L^p$ satisfying $\displaystyle \lim\limits_{n\to\infty}\|f_n\|_p=\|f\|_p$ where $\displaystyle f\in L^p$, do we necessarily have $\displaystyle <f_n>$ converges in $\displaystyle L^p$ to f (as defined above)? If not, could you please come up with a counterexample?
Thanks!
• Jun 10th 2010, 03:33 AM
Ackbeet
There's a decent chance you might find what you're looking for here. Maybe your library has it.
• Jun 10th 2010, 04:01 AM
zzzhhh
Quote:

Originally Posted by Ackbeet
There's a decent chance you might find what you're looking for here. Maybe your library has it.

No, I don't have this book. I searched just now the contents of this book from google books and the item 40 "sequences of functions converging in different senses" might be relevant, but I didn't find any useful information regarding my question from the pages that are allowed to read by google books.
• Jun 10th 2010, 04:29 AM
Focus
Quote:

Originally Posted by zzzhhh
If $\displaystyle <f_n>$ is a sequence in $\displaystyle L^p$ converging in $\displaystyle L^p$ to an f in $\displaystyle L^p$, that is, $\displaystyle \|f_n-f\|_p\to 0$ as $\displaystyle n\to\infty$, do we necessarily have $\displaystyle \lim\limits_{n\to\infty}\|f_n\|_p=\|f\|_p$? If not, could you please come up with a counterexample?

Look at the reverse triangle inequality,
$\displaystyle |\, ||f||-||g||\, | \leq ||f-g||$
Quote:

If $\displaystyle <f_n>$ is a sequence in $\displaystyle L^p$ satisfying $\displaystyle \lim\limits_{n\to\infty}\|f_n\|_p=\|f\|_p$ where $\displaystyle f\in L^p$, do we necessarily have $\displaystyle <f_n>$ converges in $\displaystyle L^p$ to f (as defined above)? If not, could you please come up with a counterexample?
Thanks!
Take the real line with the Lebesgue measure and let $\displaystyle f_n=\mathbf{1}_{[n,n+1]}$. They have $\displaystyle ||f_n||_1=1 \rightarrow ||\mathbf{1}_{[0,1]}||_1$ but it does not converge to that.
• Jun 10th 2010, 06:45 PM
zzzhhh
Brilliant! Thank you very much, Focus!
So $\displaystyle \lim\limits_{n\to\infty}\|f_n-f\|_p=0$ necessarily implies $\displaystyle \lim\limits_{n\to\infty}\|f_n\|_p=\|f\|_p$, but the converse does not. Sometimes the converse may be true, sometimes may not as your counterexample indicates. Only when it is known in advance that $\displaystyle f_n$ converges a.e. can the converse (convergence in $\displaystyle L^p$) necessarily hold.
• Jun 11th 2010, 05:49 AM
Ackbeet
Problem 6.16 in Royden, 3rd Edition seems relevant.
• Jun 17th 2010, 08:25 PM
zzzhhh
Quote:

Originally Posted by Ackbeet
Problem 6.16 in Royden, 3rd Edition seems relevant.

No, they are not relevant. I didn't assume convergence a.e.
• Jun 19th 2010, 07:37 AM
Focus
Quote:

Originally Posted by zzzhhh
Brilliant! Thank you very much, Focus!
So $\displaystyle \lim\limits_{n\to\infty}\|f_n-f\|_p=0$ necessarily implies $\displaystyle \lim\limits_{n\to\infty}\|f_n\|_p=\|f\|_p$, but the converse does not. Sometimes the converse may be true, sometimes may not as your counterexample indicates. Only when it is known in advance that $\displaystyle f_n$ converges a.e. can the converse (convergence in $\displaystyle L^p$) necessarily hold.

You need f to be in L^p in that case. More generally if the sequence is UI then a.e. convergence (or convergence in measure) implies L^p convergence. By UI I mean that
$\displaystyle \sup ||f_n \,\mathbf{1}_{|x|>K}|| \rightarrow 0$ as $\displaystyle K \rightarrow \infty$.