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Let $\{f_n\}$ be a sequence of functions in $L^p( [0,1])$, $1 \leq p < \infty$, which converges almost everywhere to a function $f$ in $L^p$. Show that $\{f_n\}$ converges to $f$ in $L^p$ norm if and only if $\|f_n\| \to \|f\|$.

What does $\|f_n\| \to \|f\|$ mean and which convergence theorem is this? I know $\{f_n\}$ converges to $f$ in $L^p$ norm means

$$ \lim_{n \to \infty} \|f_n - f\|_p = 0 $$

Mohsen Shahriari
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user770687
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    Exercise 4.17 in Brezis' text "Functional Analysis, Sobolev Spaces and Partial Differential Equations" walks you through this if you are interested. Also, $|f_{n}|{p}\to |f|{p}$ is usually referred to convergence of norm which deals with the convergence of the sequence of norms $(|f_{n}|{p}){n\geq 1}$, whereas $|f_{n}-f|_{p}\to 0$ is referred to convergence in norm. The latter is a stronger statement as it clearly implies the former, where as the converse is not true. – Zeta-Squared Aug 16 '20 at 08:29
  • Thanks @OliverDiaz that is helpful – user770687 Aug 17 '20 at 21:41

2 Answers2

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It means that $\|f_n\|_p \to \|f\|_p$ in $\Bbb R$.

FormulaWriter
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ureui
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Hints for a proof: Let $\|f_n\|_p \to \|f\|_p$. Now $2^{p}(|f|^{p}+|f_n|^{p} -|f_n-f|^{p})$ is non-negative and an application of Fatou's Lemma to this gives $\lim \sup \int |f_n-f|^{p} =0$. The converse part is true for any norm.

Jochen
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Kavi Rama Murthy
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  • Fatou's lemma tells us $$2 \int |f|^p \le \lim \inf \int (|f_n|^p + |f|^p - |f_n-f|^p).$$ But where do you go from here? $\lim \inf$ is super-additive, so we can split things up nicely. Presumably, we want to rearrange the inequality in Fatou's lemma to get something like $$\lim \sup |f_n-f|^p \le \lim \inf |f_n|^p - |f|^p.$$ But I don't see how to rearrange the inequality. – user193319 Sep 15 '21 at 15:51
  • It is given that $|f_n| \to |f|$ So $\lim \inf \int |f_n|^{p} =\lim \int |f_n|^{p} =\int |f|^{p}$. – Kavi Rama Murthy Sep 15 '21 at 23:13
  • Yes, I know that that holds true. But how do you split up $\lim \inf \int (|f_n|^p + |f|^p - |f_n-f|^p)$? – user193319 Sep 16 '21 at 15:55