Ask for two counterexamples regarding convergence in L^p

**zzzhhh** · June 10th 2010, 03:30 AM

If $<f_n>$ is a sequence in $L^p$ converging in $L^p$ to an f in $L^p$ , that is, $\|f_n-f\|_p\to 0$ as $n\to\infty$ , do we necessarily have $\lim\limits_{n\to\infty}\|f_n\|_p=\|f\|_p$ ? If not, could you please come up with a counterexample?
If $<f_n>$ is a sequence in $L^p$ satisfying $\lim\limits_{n\to\infty}\|f_n\|_p=\|f\|_p$ where $f\in L^p$ , do we necessarily have $<f_n>$ converges in $L^p$ to f (as defined above)? If not, could you please come up with a counterexample?
Thanks!

**Ackbeet** · June 10th 2010, 03:33 AM

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

**zzzhhh** · June 10th 2010, 04:01 AM

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.

**Focus** · June 10th 2010, 04:29 AM

Originally Posted by zzzhhh

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

Look at the reverse triangle inequality,
$<br /> |\, ||f||-||g||\, | \leq ||f-g||<br />$

If $<f_n>$ is a sequence in $L^p$ satisfying $\lim\limits_{n\to\infty}\|f_n\|_p=\|f\|_p$ where $f\in L^p$ , do we necessarily have $<f_n>$ converges in $L^p$ 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 $f_n=\mathbf{1}_{[n,n+1]}$ . They have $||f_n||_1=1 \rightarrow ||\mathbf{1}_{[0,1]}||_1$ but it does not converge to that.

**zzzhhh** · June 10th 2010, 06:45 PM

Brilliant! Thank you very much, Focus!
So $\lim\limits_{n\to\infty}\|f_n-f\|_p=0$ necessarily implies $\lim\limits_{n\to\infty}\|f_n\|_p=\|f\|_p$ , 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 $f_n$ converges a.e. can the converse (convergence in $L^p$ ) necessarily hold.

**Ackbeet** · June 11th 2010, 05:49 AM

Problem 6.16 in Royden, 3rd Edition seems relevant.

**zzzhhh** · June 17th 2010, 08:25 PM

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.

**Focus** · June 19th 2010, 07:37 AM

Originally Posted by zzzhhh

Brilliant! Thank you very much, Focus!
So $\lim\limits_{n\to\infty}\|f_n-f\|_p=0$ necessarily implies $\lim\limits_{n\to\infty}\|f_n\|_p=\|f\|_p$ , 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 $f_n$ converges a.e. can the converse (convergence in $L^p$ ) 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
$\sup ||f_n \,\mathbf{1}_{|x|>K}|| \rightarrow 0$ as $K \rightarrow \infty$ .

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