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Suppose $(X,M,\mu)$ is a measure space, $\mu$ a positive measure $f_n,f\in L^{p}(X)$ and $1\leq p < \infty$. I want to prove that

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

I don't really know how to prove this but I have some ideas.

By the Riesz-Fischer theorem, $\{f_n\}\to f$ in $L^p(X)$ implies that there exists a subsequence of $\{f_n\}$ that converges pointwise a.e to $f$ on $X$. And there is another theorem in Royden's which states that:

Theorem 7 Let $E$ be a measurable set and $1\leq p<\infty$. Suppose $\{f_n\}$ is a sequence in $L^p(E)$ that converges pointwise a.e. on $E$ to the function $f$ which belongs to $L^p(E)$. Then $$\{f_n\}\to f\ \text{in}\ L^p(E)\ \text{if and only}\ \lim_{n\to\infty}\int_E|f_n|^p=\int_E|f|^p.$$

Does this imply the statement is true? Any help is appreciated. Thank you!

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For any normed vector space $(V,\|\cdot\|)$, it follows from the triangle inequality that $$ \lvert\|x\|-\|y\|\rvert\leq \|x-y\| $$ for all $x,y\in V$.

Applying this inequality to $L^p(X)$, we have $$ \lvert \|f_n\|_p-\|f\|_p\rvert\leq \|f_n-f\|_p$$ hence if $f_n\to f$ in $L^p$ then $\|f_n\|_p\to \|f\|_p$.

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  • $\begingroup$ thanks! this looks very simple. But is my answer wrong? $\endgroup$
    – Heisenberg
    Jul 11 '17 at 5:06
  • $\begingroup$ The theorem in Royden proves a stronger result (if and only if) because it assumes pointwise convergence. For the direction you want, pointwise convergence is not needed. $\endgroup$ Jul 11 '17 at 5:09
  • $\begingroup$ But we get pointwise convergence from R-F $\endgroup$
    – Heisenberg
    Jul 11 '17 at 5:11
  • $\begingroup$ Or rather p.w convergence of a subsequence. Nevermind. Thank you for your help $\endgroup$
    – Heisenberg
    Jul 11 '17 at 5:12
  • $\begingroup$ Well, then using Royden's theorem as a black box would only give you the convergence you want for a subsequence. In any case, my guess is that Royden's proof of the direction you want is the same as my proof. It's actually the other direction that's more interesting. $\endgroup$ Jul 11 '17 at 5:14
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Hint: In any normed linear space $|\|x\|-\|y\|| \le \|x-y\|.$

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We are dealing the Normed linear space. So, please remember this method $||f_{n}||_{p}=||f_{n}-f+f||_{p}≤||f_{n}-f||_{p}+||f||_{p}.$

$\therefore$$||f_{n}||_{p}-||f||_{p}≤||f_{n}-f||_{p}.$

By the symmetry, $| ||f_{n}||_{p}-||f||_{p}|≤||f_{n}-f||_{p}.$ We can deduce from this.

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