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Let $1\leq p < \infty$. Suppose that

  1. $\{f_k\} \subset L^p$ (the domain here does not necessarily have to be finite),
  2. $f_k \to f$ almost everywhere, and
  3. $\|f_k\|_{L^p} \to \|f\|_{L^p}$.

Why is it the case that $\|f_k - f\|_{L^p} \to 0?$

A statement in the other direction (i.e. $\|f_k - f\|_{L^p} \to 0 \Rightarrow \|f_k\|_{L^p} \to \|f\|_{L^p}$ ) follows pretty easily and is the one that I've seen most of the time. I'm not how to show the result above though.

  • 0
    Someone would say *going to intinity* is NOT of the concept of *convergence*. But I say how if in this question $||f||_p=\infty$? In this way how can you redefine the convergence of the hypothesis of the OP's problem ?2015-11-24

2 Answers 2

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This is a theorem by Riesz.

Observe that $|f_k - f|^p \leq 2^p (|f_k|^p + |f|^p),$

Now we can apply Fatou's lemma to $2^p (|f_k|^p + |f|^p) - |f_k - f|^p \geq 0.$

If you look well enough you will notice that this implies that

$\limsup_{k \to \infty} \int |f_k - f|^p \, d\mu = 0.$

Hence you can conclude the same for the normal limit.

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    @anegligibleperson It is also a "weaker" version of parallelogram law on Banach space. There are many variants in this paper: https://pdfs.semanticscholar.org/324b/afba4fb564de640bc00ec644dcdc29d5dd02.pdf2018-08-16
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Consider $g_k = 2^p(|f_k|^p + |f|^p) - |f_k - f|^p$.

Since $g_k \geq 0$ (why?), and $g_k \to 2^{p+1}|f|^p$ a.e., we can apply Fatou's Lemma: $\int \liminf g_k \leq \liminf \int g_k$ so that $\int 2^{p+1}|f|^p \leq \liminf \left(\int 2^p |f_k|^p + \int 2^p |f|^p - \int |f_k - f|^p \right),$ and I'll let you take it from here.

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    @FardadPouran: I think that in writing $\Vert f_k \Vert_p \to \Vert f \Vert_p$, we are implicitly assuming that \Vert f \Vert_p < \infty.2015-11-25