6
$\begingroup$

I know that if $p\geq1$, $\|f_n-f\|_p\rightarrow 0$ implies $\|f_n\|_p\rightarrow \|f\|_p$, since then Minkowski inequality holds for the $L^p$ norm ($p\geq1$). Is there an example for $\|f_n-f\|_p\rightarrow 0$, but $\|f_n\|_p\nrightarrow \|f\|_p$, when $0<p<1$ and $\|f_n\|_p, \|f\|_p<\infty$?

Here $\|f\|_p=(\int_\Omega|f(x)|^p \mathrm{d}x)^{1/p}$.

Nathanael Skrepek's anwer reminded me that though the Minkowski inequality does not hold when $p<1$, the Cr inequality still implies the convergence.

$\endgroup$
4
  • $\begingroup$ Please mention what measure space we're on. $\endgroup$
    – zhw.
    May 20, 2017 at 16:21
  • $\begingroup$ @zhw. Choose one yourself for a counterexample! $\endgroup$
    – md2perpe
    May 20, 2017 at 16:31
  • $\begingroup$ In Folland there is an exercise (6.10 in my edition) reading "[Prove:] If $f_n, f \in L^p \quad (p<\infty)$ and $f_n \to f$ a.e., then $\|f_n - f\|_p \to 0$ iff $\|f_n\|_p \to \|f\|_p$." Thus we must find a sequence of functions not converging almost everywhere. $\endgroup$
    – md2perpe
    May 20, 2017 at 16:50
  • $\begingroup$ Where did you get this problem? Since such an example is impossible, you have sent some readers on a wild goose chase. Much better to ask "Is there an example ...? $\endgroup$
    – zhw.
    May 21, 2017 at 0:37

1 Answer 1

8
$\begingroup$

So I am afraid that this is not possible.

Lets denote $\rho(f) := \|f\|_p^p = \int_\Omega |f|^p \mathrm{d}\mu$. Note that $\rho$ fulfils the triangle inequality (Wikipedia reference). This leads to \begin{align}\rho(g) &= \rho\big(f+(g-f)\big) \leq \rho(f) + \rho(g-f) \\ \rho(g) - \rho(f) &\leq \rho(g-f) \end{align} and furthermore to $|\rho(g) - \rho(f) |\leq \rho(g-f)$.

Endowed with this knowledge lets consider your case: We know that from $\|f_n-f\|_p \to 0$ follows \begin{align} \rho(f_n-g)=\|f_n-f\|_p^p \to 0. \end{align} By applying the previous result we receive \begin{align} \big|\|f_n\|_p^p-\|f\|_p^p\big|=|\rho(f_n)-\rho(f)| \leq\rho(f_n-g)\to 0. \end{align} Now we know that $\|f_n\|_p^p \to \|f\|_p^p$ holds. Since the range of $\rho:f \mapsto \int_\Omega|f|^p\mathrm{d}\mu$ is $[0,+\infty)$ the function $x\mapsto x^{q}$ (for every $q>0$) is well defined and injective on the range of $\rho$. Hence we also can conclude that \begin{align} \|f_n\|_p \to \|f\|_p \end{align} holds.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .