I've seen this statement so I'm not wanting to argue it, but am thus curious as to where my logic falls short.

Folland 6.2 problem

If $1<p<\infty, \;f_n\rightarrow f$ weakly in $l^p(A)$ iff sup$_n||f_n||_p<\infty$ and $f_n\rightarrow f$ pointwise.

The classic example for $\{f_n\}\nrightarrow f $ pointwise but $\{f_n\}\rightarrow f$ in $L^p$ strongly is $\{f_n\}=\{\chi_{[0,.5]},\chi_{[0,.25]},\chi_{[.25,.5]},\chi_{[0,.125]},\chi_{[.125,.25]},...\}$ and so on in such a fashion w/ indicator functions of length $\frac{1}{2}$ to a power.
It is easy to show that this sequence does not converge pointwise but does converge in $L^p$ strongly for $1\le p<\infty$. If the sequence converges strongly then it must also converge weakly, so this sequence has to converge weakly but not pointwise inside $L^p\text{ for }1<p<\infty.$

Obviously my logic has to be wrong but where. Thanks for the help.

  • $\begingroup$ What is $A$? A subset of the natural numbers with counting measure? $\endgroup$ – Davide Giraudo Jan 2 '14 at 22:36
  • $\begingroup$ nothing like that. the context of the chapter leads me to be believe it would be any subset of $\mathbb{R}^n$. $\endgroup$ – f00d Jan 2 '14 at 22:47
  • $\begingroup$ In this case this may be not true: consider $p=2$, the unit interval and $f_n(x):=\sin(\pi n x)$. We have the weak convergence to $0$ but no pointwise convergence. $\endgroup$ – Davide Giraudo Jan 2 '14 at 22:49
  • $\begingroup$ You wrote $l^p$. That usually means $L^p$ on a set with counting measure. If you did not mistype the question, your example does not apply. $\endgroup$ – Harald Hanche-Olsen Jan 2 '14 at 22:50
  • $\begingroup$ oh right right. nevermind then, I realize what $l^p$ is, I was just staring at certain texts for too long. nevermind then. $\endgroup$ – f00d Jan 2 '14 at 23:01

The question was answered in the comments: Folland's statement is for $\ell^p$, the example is for $L^p$. I'll add the reason for the difference. The underlying measure space of $\ell^p$ is discrete, with each point having positive measure. This allows us to test weak convergence against characteristic functions of singletons. By the definition of weak convergence, the integrals $\int f_n\chi_{\{p\}}$ must converge, which implies pointwise convergence of $f_n$.

For $L^p$, the underlying measure space contains no atoms, and the above reasoning does not apply.


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